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Data Analysis And Decision Making 4th Edition By S. Christian Albright – Test Bank

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Data Analysis And Decision Making 4th Edition By S. Christian Albright – Test Bank

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CHAPTER 2: Describing the Distribution of a Single Variable

 

MULTIPLE CHOICE

 

  1. A sample of a population taken at one particular point in time is categorized as:
a. categorical c. cross-sectional
b. discrete d. time-series

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. If data is stored in a database package, which of the following terms are typically used?
a. Fields and records c. Variables and samples
b. Cases and columns d. Variables and observations

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Researchers may gain insight into the characteristics of a population by examining a
a. mathematical model describing the population
b. sample of the population
c. description of the population
d. replica

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Numerical variables can be subdivided into which two types?
a. Diverse and categorical c. Nominal and progressive
b. Discrete and continuous d. Cross-sectional and discrete

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Gender and State are examples of which type of data?
a. Discrete data c. Categorical data
b. Continuous data d. Ordinal data

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Which of the following indicates how many observations fall into various categories?
a. The Likert scale c. The sample table
b. The frequency table d. The tabulation scale

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Data that arise from counts are called:
a. continuous data c. counted data
b. nominal data d. discrete data

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A histogram that is positively skewed is also called
a. skewed to the right c. balanced
b. skewed to the left d. symmetric

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A histogram that has exactly two peaks is called a
a. unimodal distribution c. skewed distribution
b. bimodal distribution d. scatterplot

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A histogram that has a single peak and looks approximately the same to the left and right of the peak is called:
a. bimodal c. balanced
b. symmetric d. proportional

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A variable is classified as ordinal if:
a. there is a natural ordering of categories
b. there is no natural ordering of categories
c. the data arise from continuous measurements
d. we track the variable through a period of time

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. In order for the characteristics of a sample to be generalized to the entire population, it should be:
a. symbolic of the population c. representative of the population
b. typical of the population d. illustrative of the population

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. When we look at a time series plot, we usually look for which two things?
a. “Is there an observable trend?” and “Is there a seasonal pattern?”
b. “Is there an observable trend” and “Can we make predictions?”
c. “Is the sample representative?” and “Is there a seasonal pattern?”
d. “Is there an observable trend?” and “Is the trend symmetric?”

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Which of the following are possible categorizations of data type?
a. Numerical versus categorical (with subcategories nominal, ordinal)
b. Discrete versus continuous
c. Cross-sectional versus time series
d. All of these options
e. Two of these options

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Which of the following are the two most commonly used measures of variability?
a. Variance and median
b. Variance and standard deviation
c. Mean and variance
d. Mean and range
e. First quartile and third quartile

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The median can also be described as:
a. the middle observation when the data values are arranged in ascending order
b. the second quartile
c. the 50th percentile
d. All of these options

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The difference between the first and third quartile is called the
a. interquartile range
b. interdependent range
c. unimodal range
d. bimodal range
e. mid range

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. If a value represents the 95th percentile, this means that
a. 95% of all values are below this value
b. 95% of all values are above this value
c. 95% of the time you will observe this value
d. there is a 5% chance that this value is incorrect
e. there is a 95% chance that this value is correct

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. For a boxplot, the point inside the box indicates the location of the
a. mean c. minimum value
b. median d. maximum value

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. For a boxplot, the vertical line inside the box indicates the location of the
a. mean
b. median
c. mode
d. minimum value
e. maximum value

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Which of the following are the three most common measures of central location?
a. Mean, median, and mode
b. Mean, variance, and standard deviation
c. Mean, median, and variance
d. Mean, median, and standard deviation
e. First quartile, second quartile, and third quartile

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The length of the box in the boxplot portrays the
a. mean
b. median
c. range
d. interquartile range
e. third quartile

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Suppose that a histogram of a data set is approximately symmetric and “bell shaped”. Approximately what percent of the observations are within two standard deviations of the mean?
a. 50%
b. 68%
c. 95%
d. 99.7%
e. 100%

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. The mode is best described as the
a. middle observation
b. same as the average
c. 50th percentile
d. most frequently occurring value
e. third quartile

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. For a boxplot, the box itself represents what percent of the observations?
a. lower 25%
b. middle 50%
c. upper 75%
d. upper 90%
e. 100%

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Which of the following statements is true for the following data values: 7, 5, 6, 4, 7, 8, and 12?
a. The mean, median and mode are all equal
b. Only the mean and median are equal
c. Only the mean and mode are equal
d. Only the median and mode are equal

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. In a histogram, the percentage of the total area which must be to the left of the median is:
a. exactly 50%
b. less than 50% if the distribution is skewed to the left
c. more than 50% if the distribution is skewed to the right
d. between 25% and 50% if the distribution is symmetric and unimodal

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The average score for a class of 30 students was 75. The 20 male students in the class averaged 70. The 10 female students in the class averaged:
a. 75
b. 85
c. 60
d. 70
e. 80

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Which of the following statements is true?
a. The sum of the deviations from the mean is always zero
b. The sum of the squared deviations from the mean is always zero
c. The range is always smaller than the variance
d. The standard deviation is always smaller than the variance

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Expressed in percentiles, the interquartile range is the difference between the
a. 10th and 60th percentiles
b. 15th and 65th percentiles
c. 20th and 70th percentiles
d. 25th and 75th percentiles
e. 35th and 85th percentiles

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. A sample of 20 observations has a standard deviation of 4. The sum of the squared deviations from the sample mean is:
a. 400
b. 320
c. 304
d. 288
e. 180

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

TRUE/FALSE

 

  1. Age, height, and weight are examples of numerical data.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Data can be categorized as cross-sectional or time series.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. All nominal data may be treated as ordinal data.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Four different shapes of histograms are commonly observed: symmetric, positively skewed, negatively skewed, and bimodal.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Categorical variables can be classified as either discrete or continuous.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A skewed histogram is one with a long tail extending either to the right or left. The former is called negatively skewed, and the later is called positively skewed.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Some histograms have two or more peaks. This is often an indication that the data come from two or more distinct populations.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. A population includes all elements or objects of interest in a study, whereas a sample is a subset of the population used to gain insights into the characteristics of the population.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. A frequency table indicates how many observations fall within each category, and a histogram is its graphical analog.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. In the term “frequency table,” frequency refers to the number of data values falling within each category.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. The number of car insurance policy holders is an example of a discrete random variable

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A variable (or field) is an attribute, or measurement, on members of a population, whereas an observation (or case or record) is a list of all variable values for a single member of a population.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Phone numbers, Social Security numbers, and zip codes are examples of numerical variables.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Cross-sectional data are data on a population at a distinct point in time, whereas time series data are data collected across time.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Distribution is a general term used to describe the way data are distributed, as indicated by a frequency table or histogram.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Both ordinal and nominal variables are categorical.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. A histogram is said to be symmetric if it has a single peak and looks approximately the same to the left and right of the peak.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic

 

  1. Suppose that a sample of 10 observations has a standard deviation of 3, then the sum of the squared deviations from the sample mean is 30.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. If a histogram has a single peak and looks approximately the same to the left and right of the peak, we should expect no difference in the values of the mean, median, and mode.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The mean is a measure of central location.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The length of the box in the boxplot portrays the interquartile range.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. In a positively skewed distribution, the mean is smaller than the median and the median is smaller than the mode.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The value of the standard deviation always exceeds that of the variance.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The difference between the first and third quartiles is called the interquartile range.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The standard deviation is measured in original units, such as dollars and pounds.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The median is one of the most frequently used measures of variability.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Assume that the histogram of a data set is symmetric and bell shaped, with a mean of 75 and standard deviation of 10. Then, approximately 95% of the data values were between 55 and 95.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Abby has been keeping track of what she spends to rent movies. The last seven week’s expenditures, in dollars, were 6, 4, 8, 9, 6, 12, and 4. The mean amount Abby spends on renting movies is $7.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Expressed in percentiles, the interquartile range is the difference between the 25th and 75th percentiles.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The value of the mean times the number of observations equals the sum of all of the data values.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The difference between the largest and smallest values in a data set is called the range.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. There are four quartiles that divide the values in a data set into four equal parts.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Suppose that a sample of 8 observations has a standard deviation of 2.50, then the sum of the squared deviations from the sample mean is 17.50.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The median of a data set with 30 values would be the average of the 15th and the 16th values when the data values are arranged in ascending order.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

SHORT ANSWER

 

NARRBEGIN: SA_71_73

 

A manager for Marko Manufacturing, Inc. has recently been hearing some complaints that women are being paid less than men for the same type of work in one of their manufacturing plants. The boxplots shown below represent the annual salaries for all salaried workers in that facility (40 men and 34 women).

 

NARREND

 

 

  1. Would you conclude that there is a difference between the salaries of women and men in this plant? Justify your answer.

 

ANS:

Yes. The men seem to have higher salaries than the women do in many cases. We can see from the boxplots that the mean and median values for the men are both higher than for the women. You can also see from the boxplots that the middle 50% of salaries for men is above the median for women. This means that if you were in the 25th percentile for men, you would be above the 50th percentile for women. You can also see that the mean and median salaries for the men are about $10,000 above those for the women.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. How large must a person’s salary should be to qualify as an outlier on the high side? How many outliers are there in these data?

 

ANS:

A person’s salary should be somewhere above $70,000. There is one male salary that would be considered an outlier (at approximately $80,000)

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. What can you say about the shape of the distributions given the boxplots above?

 

ANS:

 

They both appear to be slightly skewed to the right (both have a mean > median). The total variation seems to be close for both distributions (with one outlier for the male salaries), but there seems to be more variation in the middle 50% for the women than for the men. There seem to be more men’s salaries clustered more closely around the mean than for the women.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_74_75

Statistics professor has just given a final examination in his statistical inference course. He is particularly interested in learning how his class of 40 students performed on this exam. The scores are shown below.

 

77        81        74        77        79        73        80        85        86        73

83        84        81        73        75        91        76        77        95        76

90        85        92        84        81        64        75        90        78        78

82        78        86        86        82        70        76        78        72        93

NARREND

 

 

  1. What are the mean and median scores on this exam?

 

ANS:

 

Mean = 80.40, Median = 79.50

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Explain why the mean and median are different.

 

ANS:

 

There are few higher exam scores that tend to pull the mean away from the middle of the distribution. While there is a slight amount of positive skewness in the distribution (skewness = 0.182), the mean and the median are essentially equivalent in this case.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

NARRBEGIN: SA_76_78

The data shown below contains family incomes (in thousands of dollars) for a set of 50 families; sampled in 1980 and 1990. Assume that these families are good representatives of the entire United States.

1980 1990 1980 1990 1980 1990
 58  54  33  29  73  69
  6   2  14  10  26  22
 59  55  48  44  64  70
 71  57  20  16  59  55
 30  26  24  20  11  7
 38  34  82  78  70  66
 36  32  95  97  31  27
 33  29  12   8  92  88
 72  68  93  89 115 111
100  96 100 102  62  58
 1   0  51  47  23  19
 27  23  22  18  34  30
 22  47  50  75  36  61
141 166 124 149 125 150
 72  97 113 138 121 146
165 190 118 143  88 113
 79 104  96 121    

 

 

NARREND

 

 

  1. Find the mean, median, standard deviation, first and third quartiles, and the 95th percentile for family incomes in both years.

 

ANS:

Income 1980   Income 1990

Mean

Median

Standard deviation

First quartile

Third quartile

95th percentile

62.820

59.000

39.786

30.250

92.750

124.550

67.120

57.500

48.087

27.500

97.000

149.55

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The Republicans claim that the country was better off in 1990 than in 1980, because the average income increased. Do you agree?

 

ANS:

 

It is true that the mean increased slightly, but the median decreased and the standard deviation increased. The 95th percentile shows that the mean increase might be because the rich got richer.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Generate a boxplot to summarize the data. What does the boxplot indicate?

 

ANS:

The boxplot shows that there is not much difference between the two populations.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_79_81

In an effort to provide more consistent customer service, the manager of a local fast-food restaurant would like to know the dispersion of customer service times about their average value for the facility’s drive-up window. The table below provides summary measures for the customer service times (in minutes) for a sample of 50 customers collected over the past week.

 

Count 50.000
Mean 0.873
Median 0.885
Standard deviation 0.432
Minimum 0.077
Maximum 1.608
Variance 0.187
Skewness -0.003

 

NARREND

 

 

  1. Interpret the variance and standard deviation of this sample.

 

ANS:

 

The variance = 0.187 (minutes squared) and this represents the average of the squared deviations from the mean. The standard deviation = 0.432 (minutes) and is the square root of the variance. Both the variance and standard deviation measure the variation around the mean of the data. However, it is easier to interpret the standard deviation because it is expressed in the same units (minutes) as the values of the random variable (customer service time).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Are the empirical rule applicable in this case? If so, apply it and interpret your results. If not, explain why the empirical rule is not applicable here.

 

ANS:

 

Considering that this distribution is only very slightly skewed to the left, it is acceptable to apply the empirical rule as follows:

Approximately 68% of the customer service times will fall between 0.873 ± 0.432, that is between 0.441 and 1.305 minutes.

Approximately 95% of the customer service times will fall between 0.873 ± 2(0.432), that is between 0.009 and 1.737 minutes.

Approximately 99.7% of the customer service times will fall between 0.873 ± 3(0.432), that is between 0 and 2.169 (we set the lower end to zero since service times cannot assume negative values).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Explain what would cause the mean to be slightly lower than the median in this case.

 

ANS:

 

The data is slightly skewed to the left. This is causing the mean to be slightly lower than the median. It is important to understand that service times are bounded on the lower end by zero (or it is impossible for the service time to be negative). However, there is no bound on the maximum service time. Therefore, the smaller service times are causing the mean to be somewhat lower than the median.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_82_85

Below you will find summary measures on salaries for classroom teachers across the United States. You will also find a list of selected states and their average teacher salary. All values are in thousands of dollars.

 

Salaries for classroom teachers across the United States

  Salary
Count 51.000
Mean 35.890
Median 35.000
Standard deviation  6.226
Minimum 26.300
Maximum 50.300
Variance 38.763
First quartile 31.550
Third quartile 40.050

 

Selected states and their average teacher salary

State Salary
Alabama 31.3
Colorado 35.4
Connecticut 50.3
Delaware 40.5
Nebraska 31.5
Nevada 36.2
New Hampshire 35.8
New Jersey 47.9
New Mexico 29.6
South Carolina 31.6
South Dakota 26.3
Tennessee 33.1
Texas 32.0
Utah 30.6
Vermont 36.3
Virginia 35.0
Wyoming 31.6

 

NARREND

 

 

  1. Which of the states listed paid their teachers average salaries that exceed at least 75% of all average salaries?

 

ANS:

 

Connecticut at 50.3; Delaware at 40.5; and New Jersey at 47.9 (all those > 40.05).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Which of the states listed paid their teachers average salaries that are below 75% of all average salaries?

 

ANS:

 

Alabama at 31.3; Nebraska at 31.5; New Mexico at 29.6; South Dakota at 26.3; and Utah at 30.6 (all those < 31.55).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. What salary amount represents the second quartile?

 

ANS:

 

$35,000 (median)

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. How would you describe the salary of Virginia’s teachers compared to those across the entire United States? Justify your answer.

 

ANS:

 

Virginia = $35,000 which is also the median. Virginia is at the 50th percentile or 50% of the teachers’ salaries across the U.S. are below Virginia and 50% of the salaries are above theirs.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_86_88

Suppose that an analysis of a set of test scores reveals that: ,

NARREND

 

 

  1. What do these statistics tell you about the shape of the distribution?

 

ANS:

 

The fact that 40 is greater that 20 indicates that the distribution is skewed to the left.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. What can you say about the relative position of each of the observations 34, 84, and 104?

 

ANS:

 

Since 34 is less than , the observation 34 is among the lowest 25% of the values. The value 84 is a bit smaller than the middle value, which is  85. Since 105, the value 104 is larger than about 75% of the values.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Calculate the interquartile range. What does this tell you about the data?

 

ANS:

 

IQR =  60. This means that the middle 50% of the test scores are between 45 and 105.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_89_91

The following data represent the number of children in a sample of 10 families from Chicago: 4, 2, 1, 1, 5, 3, 0, 1, 0, and 2.

NARREND

 

 

  1. Compute the mean number of children.

 

ANS:

 

Mean = 1.90

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Compute the median number of children.

 

ANS:

 

Median = 1.5

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Is the distribution of the number of children symmetrical or skewed? Why?

 

ANS:

 

The distribution is positively skewed because the mean is larger than the median.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. The data below represents monthly sales for two years of beanbag animals at a local retail store (Month 1 represents January and Month 12 represents December). Given the time series plot below, do you see any obvious patterns in the data? Explain.

 

 

ANS:

 

This is a representation of seasonal data. There seems to be a small increase in months 3, 4, and 5 and a large increase at the end of the year. The sales of this item seem to peak in December and have a significant drop off in January.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. An operations management professor is interested in how her students performed on her midterm exam. The histogram shown below represents the distribution of exam scores (where the maximum score is 100) for 50 students.

 

 

Based on this histogram, how would you characterize the students’ performance on this exam?

 

ANS:

 

Exam scores are fairly normally distributed. Majority of scores (76%) are between 70 and 90 points, while 12% of scores are above 90 and 12% of scores are 70 or below.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. The proportion of Americans under the age of 18 who are living below the poverty line for each of the years 1959 through 2000 is used to generate the following time series plot.

 

How successful have Americans been recently in their efforts to win “the war against poverty” for the nation’s children?

 

ANS:

 

Americans have been relatively unsuccessful in winning the war on poverty in the 1990s. This is especially true when you compare recent poverty rates with those of the years from 1969 through 1979. However, at least the curve is trending downwards in the most recent years.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_95_97

A financial analyst collected useful information for 30 employees at Gamma Technologies, Inc. These data include each selected employees gender, age, number of years of relevant work experience prior to employment at Gamma, number of years of employment at Gamma, the number of years of post-secondary education, and annual salary.

NARREND

 

 

  1. Indicate the type of data for each of the six variables included in this set.

 

ANS:

 

Gender – categorical, nominal

Age – numerical, continuous

Prior experience – numerical, discrete

Gamma experience – numerical, discrete

Education – numerical, discrete

Annual salary – numerical, continuous

 

PTS:   1                    MSC:  AACSB: Analytic

 

  1. Based on the histogram shown below, how would you describe the age distribution for these data?

 

 

ANS:

 

The age distribution is skewed slightly to the right. Largest grouping is in the 30-40 range. This means that most workers are above the age of 30 years and only one worker is 20 years old or younger.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

  1. Based on the histogram shown below, how would you describe the salary distribution for these data?

 

ANS:

 

The salary distribution is skewed to the right. There appears to be several workers who are being paid substantially more than the others. If you eliminate those above $80,000, the salaries are fairly normally distributed around $35,000.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

NARRBEGIN: SA_98_103

The histogram below represents scores achieved by 250 job applicants on a personality profile.

 

NARREND

 

 

  1. What percentage of the job applicants scored between 30 and 40?

 

ANS:

 

10%

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. What percentage of the job applicants scored below 60?

 

ANS:

 

90%

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. How many job applicants scored between 10 and 30?

 

ANS:

 

100

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. How many job applicants scored above 50?

 

ANS:

 

50

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Seventy percent of the job applicants scored above what value?

 

ANS:

 

20

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. Half of the job applicants scored below what value?

 

ANS:

 

30

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Descriptive Statistics

 

  1. A question of great interest to economists is how the distribution of family income has changed in the United States during the last 20 years. The summary measures and histograms shown below are generated for a sample of 500 family incomes, using the 1985 and 2005 income for each family in the sample.

 

Summary Measures:

 

 

 

 

 

 

Based on these results, discuss as completely as possible how the distribution of family income in the United States changed from 1985 to 2005.

 

ANS:

 

These summary measures say quite a lot. The mean has increased, although the median has decreased. There is also more variation. In fact, the 5th percentile has decreased slightly, whereas the 95th percentile is much larger — the rich people are getting richer. This behavior is also evident in the two histograms (which use the same categories for ease of comparison).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Statistical Inference

 

CHAPTER 4: Probability and Probability Distributions

 

MULTIPLE CHOICE

 

  1. Probabilities that cannot be estimated from long-run relative frequencies of events are
a. objective probabilities c. complementary probabilities
b. subjective probabilities d. joint probabilities

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The probability of an event and the probability of its complement always sum to:
a. 1 c. any value between 0 and 1
b. 0 d. any positive value

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If events A and B are mutually exclusive, then the probability of both events occurring simultaneously is equal to
a. 0.0 c. 1.0
b. 0.5 d. any value between 0.5 and 1.0

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Probabilities that can be estimated from long-run relative frequencies of events are
a. objective probabilities c. complementary probabilities
b. subjective probabilities d. joint probabilities

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Let A and B be the events of the FDA approving and rejecting a new drug to treat hypertension, respectively. The events A and B are:
a. independent c. unilateral
b. conditional d. mutually exclusive

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. A function that associates a numerical value with each possible outcome of an uncertain event is called a
a. conditional variable c. population variable
b. random variable d. sample variable

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The formal way to revise probabilities based on new information is to use:
a. complementary probabilities c. unilateral probabilities
b. conditional probabilities d. common sense probabilities

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1.  is the:
a. addition rule c. rule of complements
b. commutative rule d. rule of opposites

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The law of large numbers is relevant to the estimation of
a. objective probabilities c. both of these options
b. subjective probabilities d. neither of these options

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. A discrete probability distribution:
a. lists all of the possible values of the random variable and their corresponding probabilities
b. is a tool that can be used to incorporate uncertainty into models
c. can be estimated from long-run proportions
d. is the distribution of a single random variable

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Which of the following statements are true?
a. Probabilities must be nonnegative
b. Probabilities must be less than or equal to 1
c. The sum of all probabilities for a random variable must be equal to 1
d. All of these options are true.

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If P(A) = P(A|B), then events A and B are said to be
a. mutually exclusive c. exhaustive
b. independent d. complementary

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If A and B are mutually exclusive events with P(A) = 0.70, then P(B):
a. can be any value between 0 and 1
b. can be any value between 0 and 0.70
c. cannot be larger than 0.30
d. Cannot be determined with the information given

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are collectively exhaustive, what is the probability that one or the other occurs?
a. 0.25
b. 0.50
c. 1.00
d. Cannot be determined from the information given.

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are collectively exhaustive, what is the probability that both occur at the same time?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The joint probabilities shown in a table with two rows,  and and two columns,  and , are as follows: P( and ) = .10, P( and ) = .30, P( and ) = .05, and P(and ) = .55. Then P(|), calculated up to two decimals, is
a. .33 c. .65
b. .35 d. .67

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are mutually exclusive, what is the probability that one or the other occurs?
a. 0.25
b. 0.50
c. 1.00
d. Cannot be determined from the information given.

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are mutually exclusive, what is the probability that both occur at the same time?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are mutually exclusive and collectively exhaustive, what is the probability that both occur?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. There are two types of random variables, they are
a. discrete and continuous c. complementary and cumulative
b. exhaustive and mutually exclusive d. real and unreal

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If P(A) = 0.25 and P(B) = 0.65, then P(A and B) is:
a. 0.25
b. 0.40
c. 0.90
d. Cannot be determined from the information given

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If two events are independent, what is the probability that they both occur?
a. 0
b. 0.50
c. 1.00
d. Cannot be determined from the information given

 

 

ANS:  D                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If A and B are any two events with P(A) = .8 and P(B|) = .7, then P(and B) is
a. .56 c. .24
b. .14 d. None of the above

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Which of the following best describes the concept of marginal probability?
a. It is a measure of the likelihood that a particular event will occur, regardless of whether another event occurs.
b. It is a measure of the likelihood that a particular event will occur, given that another event has already occurred.
c. It is a measure of the likelihood of the simultaneous occurrence of two or more events.
d. None of the above.

 

 

ANS:  A                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The joint probabilities shown in a table with two rows,  and and two columns,  and , are as follows: P( and ) = .10, P( and ) = .30, P( and ) = .05, and P(and ) = .55. Then P(|), calculated up to two decimals, is
a. .33 c. .65
b. .35 d. .67

 

 

ANS:  B                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If A and B are mutually exclusive events with P(A) = 0.30 and P(B) = 0.40, then the probability that either A or B or both occur is:
a. 0.10 c. 0.70
b. 0.12 d. None of the above

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If A and B are any two events with P(A) = .8 and P(B|A) = .4, then the joint probability of A and B is
a. .80 c. .32
b. .40 d. 1.20

 

 

ANS:  C                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

TRUE/FALSE

 

  1. If A and B are independent events with P(A) = 0.40 and P(B) = 0.50, then P(A/B) is 0.50.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. A random variable is a function that associates a numerical value with each possible outcome of a random phenomenon.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two or more events are said to be exhaustive if one of them must occur.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. You think you have a 90% chance of passing your statistics class. This is an example of subjective probability.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The number of cars produced by GM during a given quarter is a continuous random variable.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two events A and B are said to be independent if P(A and B) = P(A) + P(B)

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Probability is a number between 0 and 1, inclusive, which measures the likelihood that some event will occur.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If events A and B have nonzero probabilities, then they can be both independent and mutually exclusive.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The probability that event A will not occur is denoted as .

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If P(A and B) = 1, then A and B must be collectively exhaustive.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Conditional probability is the probability that an event will occur, with no other events taken into consideration.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. When we wish to determine the probability that at least one of several events will occur, we would use the addition rule.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The law of large numbers states that subjective probabilities can be estimated based on the long run relative frequencies of events

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two events are said to be independent when knowledge of one event is of no value when assessing the probability of the other.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose A and B are mutually exclusive events where P(A) = 0.2 and P(B) = 0.5, then P(A or B) = 0.70.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If A and B are two independent events with P(A) = 0.20 and P(B) = 0.60, then P(A and B) = 0.80

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The relative frequency of an event is the number of times the event occurs out of the total number of times the random experiment is run.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Marginal probability is the probability that a given event will occur, given that another event has already occurred.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The temperature of the room in which you are writing this test is a continuous random variable.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two events A and B are said to mutually be exclusive if P(A and B) = 0.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two or more events are said to be exhaustive if at most one of them can occur.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. When two events are independent, they are also mutually exclusive.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Two or more events are said to be mutually exclusive if at most one of them can occur.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Given that events A and B are independent and that P(A) = 0.8 and P(B/A) = 0.4, then P(A and B) = 0.32.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The time students spend in a computer lab during one day is an example of a continuous random variable.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The multiplication rule for two events A and B is: P(A and B) = P(A|B)P(A).

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The number of car insurance policy holders is an example of a discrete random variable.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose A and B are mutually exclusive events where P(A) = 0.3 and P(B) = 0.4, then P(A and B) = 0.12.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose A and B are two events where P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, then P(B/A) = 0.5.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose that after graduation you will either buy a new car (event A) or take a trip to Europe (event B). Events A and B are mutually exclusive.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If P(A and B) = 0, then A and B must be collectively exhaustive.

 

ANS:  F                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. The number of people entering a shopping mall on a given day is an example of a discrete random variable.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Football teams toss a coin to see who will get their choice of kicking or receiving to begin a game. The probability that given team will win the toss three games in a row is 0.125.

 

ANS:  T                    PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

SHORT ANSWER

 

NARRBEGIN: SA_61_65

A manufacturing facility needs to open a new assembly line in four months or there will be significant cost overruns. The manager of this project believes that there are four possible values for the random variable X (the number of months from now it will take to complete this project): 3, 3.5, 4, and 4.5. It is currently believed that the probabilities of these four possibilities are in the ratio 1 to 2 to 3 to 2. That is, X = 3.5 is twice as likely as X = 3 and X = 4 is 1.5 times as likely as X = 3.5.

NARREND

 

 

  1. Find the probability distribution of X.

 

ANS:

 

x 3 3.5 4 4.5
P (X = x) 0.125 0.250 0.375 0.250

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that this project will be completed in less than 4 months from now?

 

ANS:

P(X < 4) = 0.375

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that this project will not be completed on time?

 

ANS:

P(X > 4) = 0.250

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. (A) What is the expected completion time (in months) from now for this project?

 

(B) How much variability (in months) exists around the expected value found in (A)?

 

ANS:

(A) E(X) = 3.875 months

 

(B) Var(X) = 0.2343; Stdev (X) = 0.4840 months

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_70_78

A small grocery store has two checkout lines available to its customers: a regular checkout line and an express checkout line. Customers with 5 or fewer items are expected to use the express line. Let X and Y be the number of customers in the regular checkout line and the express checkout line, respectively. Note that these numbers include the customers being served, if any. The joint probability distribution of X and Y is given in the table below.

 

  Y = 0 Y = 1 Y = 2 3
X = 0 0.06 0.04 0.03 0.15
X = 1 0.09 0.06 0.03 0.04
X = 2 0.08 0.05 0.01 0.12
3 0.07 0.05 0.03 0.09

 

NARREND

 

 

  1. Find the marginal distribution of X. What does this distribution tell you?

 

ANS:

The marginal distribution of X is:

 

This distribution indicates the likelihood of observing a particular number of customers in the regular checkout line.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Find the marginal distribution of Y. What does this distribution tell you?

 

ANS:

The marginal distribution of Y is:

This distribution indicates the likelihood of observing a particular number of customers in the express checkout line.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. (A) Calculate the conditional distribution of X given Y.

 

(B) What is the practical benefit of knowing the conditional distribution in (A)?

 

ANS:

(A) The conditional distribution of X given Y is:

 

  Y = 0 Y = 1 Y = 2 3
X = 0 0.200 0.200 0.300 0.375
X = 1 0.300 0.300 0.300 0.100
X = 2 0.267 0.250 0.100 0.300
3 0.233 0.250 0.300 0.225
  1.00 1.00 1.00 1.00

 

(B) If we find that the probability that customers are waiting in the regular line when the express line is empty is relatively large, we might permit some customers in the regular line to switch to the express line when it is empty. Conversely, if we learn that the probability that no customers are waiting in the regular line when the express line is busy is relatively large, we might then encourage express line customers to switch to the idle regular line. The idea here is to reduce the average waiting time of the customers.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Calculate the conditional distribution of Y given X.

 

ANS:

The conditional distribution of Y given X is

           

  Y = 0 Y = 1 Y = 2 Y  3 Total
X = 0 0.214 0.143 0.107 0.536 1.00
X = 1 0.409 0.273 0.136 0.182 1.00
X = 2 0.308 0.192 0.038 0.462 1.00
X  3 0.292 0.208 0.125 0.375 1.00

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that no one is waiting or being served in the regular checkout line?

 

ANS:

P(Regular line is empty) = P(X=0) = 0.28

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that no one is waiting or being served in the express checkout line?

 

ANS:

P(Express line is empty) = P(Y=0) = 0.30

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that no more than two customers are waiting in both lines combined?

 

ANS:

=

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. On average, how many customers would you expect to see in each of these two lines at the grocery store?

 

ANS:

Expected number of customers in regular line = E(X) = 1.46

Expected number of customers in express line = E(Y) = 1.60

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_79_83

Suppose that the manufacturer of a particular product assesses the joint distribution of the price per unit (P) and demand (D) for its product in the upcoming quarter as presented below. Use this information to answer the following questions.

 

Demand (D)

Price per Unit (P) 2000 2500 3000 3500  
$20 0.05 0.05 0.03 0.15 0.28
$25 0.05 0.06 0.10 0.05 0.26
$30 0.08 0.10 0.04 0.03 0.25
$35 0.10 0.05 0.03 0.03 0.21
  0.28 0.26 0.20 0.26  

 

NARREND

 

 

  1. Find the expected price and demand level for the upcoming quarter.

 

ANS:

E(P) = $26.95; E(D) = 2720 units

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that the price of this product will be above its mean in the upcoming quarter?

 

ANS:

P(P > 26.95) = 0.25 + 0.21 = 0.46

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that the demand of this product will be below its mean in the upcoming quarter?

 

ANS:

P(D < 2720) = 0.28 + 0.26 = 0.54

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that the demand of this product exceed 2500 units in the upcoming quarter, given that its price will be less than $30?

 

ANS:

P(D > 2500|P<30)=(.03 + .15 + .10 + .05)/(.28 + .26) = .33/.54 = .6111

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that the demand of this product will be less than 3500 units in the upcoming quarter, given that its price will be greater than $20?

 

ANS:

P(D < 3500|P > 20) = (.05 + .06 + .10 + .08 + .10 + .04 + .10 + .05 + .03)/(.26 + .25 + .21)

        = .61/.72 = .8472

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_84_90

A sporting goods store sells two competing brands of softball bats. Let  and  be the numbers of the two brands sold on a typical day at the store. Based on the store historical data, the conditional probability distribution of  given  is assessed and provided in the table below. The marginal distribution of  is also given in the bottom row of the table.

 

Sales of Brand 1, Given sales of Brand 2

 

   

= 0

 

= 1

 

= 2

 

= 3

= 0 0.05 0.15 0.25 0.30
= 1 0.10 0.25 0.55 0.57
= 2 0.60 0.50 0.15 0.10
= 3 0.25 0.10 0.05 0.03
Marginal Distribution of 0.20 0.30 0.30 0.20

 

NARREND

 

 

  1. Areand independent random variables? Explain why or why not.

 

ANS:

No. The, this means that given that  has occurred, this changes the probability of X1 occurring. Or, you can also say that selling one type of bat (e.g., ) reduces the probability of selling another brand of bat (e.g., ).

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Calculate the joint probabilities of  and .

 

ANS:

The formula P() = P() P() is used to generate the joint probability of  and .

 

 

Sales of Brand 2

Sales of Brand 1 = 0 = 1 = 2 = 3
= 0 0.01 0.045 0.075 0.060
 = 1 0.02 0.075 0.165 0.114
 = 2 0.12 0.150 0.045 0.020
 = 3 0.05 0.03 0.015 0.006

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Determine the marginal probability distribution of .

 

ANS:

P(= 0 ) = 0.190, P(= 1 ) = 0.374, P(= 2 ) = 0.335, P(= 3 ) = 0.101

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is probability of observing the sale of at least one brand 1 bat and at least one brand 2 bat on the same day at this sporting goods store?

 

ANS:

This is P( > 0 and  > 0), and can be calculated from the joint probabilities in Question 111. The answer is 0.62, which includes all probabilities for = 1, 2, 3 and  = 1, 2, 3.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability of observing the sale of at least one brand 1 bat on a given day at this sporting goods store?

 

ANS:

This is P ( > 0) or P ( = 1) + P ( = 2) + P ( = 3). The answer is 0.81.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability of observing the sale of no more than two brand 2 bats on a given day at this sporting goods store?

 

ANS:

This is P( £ 2) or P(= 2) + P( = 1) + P( = 0). The answer is 0.80.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Given that no brand 2 bats are sold on a given day, what is the probability of observing the sale of at least one brand 1 bicycle at this sporting goods store?

 

ANS:

This is P( ³ 1 | = 0), which can be found in column 1 ( = 0) of the original table. The answer is 0.95.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_91_103

A sample of 1000 households was selected in Los Angeles to determine information concerning consumer behavior. Among the questions asked was “Do you enjoy shopping for clothing?” Of 480 males, 272 answered yes. Of 520 females, 448 answered yes.

NARREND

 

 

  1. Set up a 22 contingency table for this situation.

 

ANS:

Gender

Enjoy Shopping for Clothing Male Female Total
Yes 272 448 720
No 208 72 280
Total 480 520 1000

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Give an example of a simple event.

 

ANS:

Since simple events have only one criterion specified, an example could be any one of the following: being a male, being a female, enjoying clothes shopping, not enjoying clothes shopping.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Give an example of a joint event.

 

ANS:

Since joint events specify two criteria simultaneously, an example could be any one of the following: being a male and enjoying clothes shopping, being a male and not enjoying clothes shopping, being a female and enjoying clothes shopping, being a female and not enjoying clothes shopping.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a male?

 

ANS:

P(male) = 480/1000 = 0.48

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random enjoys shopping for clothing?

 

ANS:

P(enjoys shopping for clothing) = 720/1000 = 0.72

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a male and enjoys shopping for clothing?

 

ANS:

P(male and enjoys shopping for clothing) = 272/1000 = 0.272

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a female and enjoys shopping for clothing?

 

ANS:

P(female and enjoys shopping for clothing) = 448/1000 = 0.448

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a male and does not enjoy shopping for clothing?

 

ANS:

P(male and does not enjoy shopping for clothing) = 208/1000 = 0.208

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a female or enjoys shopping for clothing?

 

ANS:

P(female or enjoys clothes shopping) = (520+720 – 448) /1000 = 792/1000 = 0.792

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a male or does not enjoy shopping for clothing?

 

ANS:

P(male or does not enjoy clothes shopping) = (480+280-208)/1000 = 552/1000 = 0.552

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random is a male or a female?

 

ANS:

P(male or female) = (480 + 520) / 1000 = 1000/1000 = 1.00

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that a respondent chosen at random enjoys or does not enjoy shopping for clothing?

 

ANS:

P(enjoys or does not enjoy shopping for clothing) = (720 + 280) / 1000 = 1.00

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Does consumer behavior depend on the gender of consumer? Explain using probabilities.

 

ANS:

P(male and enjoys shopping for clothing) = 0.272

P(male) . P(enjoys shopping for clothing) = (0.48)(0.72) = 0.3456

Since 0.272 0.3456, we conclude that consumer behavior and gender are dependent of each other.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_104_113

Suppose that patrons of a restaurant were asked whether they preferred beer or whether they preferred wine. 60% said that they preferred beer. 70% of the patrons were male. 80% of the males preferred beer.

NARREND

 

 

  1. Construct the joint probability table.

 

ANS:

M = Male, F = Female, B = Beer, W = Wine

P(M) = .70, P(B) = .60, P(B/M) = .80 P(B and M) = .56.

           

The joint probability table is shown below.

 

Drinking Preference

Gender B W Total
M 0.56 0.14 0.70
F 0.04 0.26 0.30
Total 0.60 0.40 1.00

 

 

PTS:   1

 

  1. What is the probability a randomly selected patron prefers wine?

 

ANS:

P(W) = 0.4

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability a randomly selected patron is a female?

 

ANS:

P(F) = 0.30

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability a randomly selected patron is a female who prefers wine?

 

ANS:

P(F and W) = 0.26

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability a randomly selected patron is a female who prefers beer?

 

ANS:

P(F and B) = 0.04

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose a randomly selected patron prefers wine. What is the probability the patron is a male?

 

ANS:

P(M|W) = 0.35

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose a randomly selected patron prefers beer. What is the probability the patron is a male?

 

ANS:

P(M|B) = 0.933

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose a randomly selected patron is a female. What is the probability the patron prefers beer?

 

ANS:

P(B|F) = 0.133

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose a randomly selected patron is a female. What is the probability that the patron prefers wine?

 

ANS:

P(W|F) = 0.867

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Are gender of patrons and drinking preference independent? Explain.

 

ANS:

P(W|F) = 0.867, and P(W) = 0.40. Since P(W|F)  P(W), we conclude that the two events are dependent. In other words, drinking preference depends on the gender of patrons.

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

NARRBEGIN: SA_113_120

An oil company is planning to drill three exploratory wells in different areas of West Texas. The company estimates that each of these wells, independent of the others, has about a 30% chance of being successful.

NARREND

 

 

  1. Find the probability distribution of X; the number of oil wells that will be successful.

 

ANS:

 

X 0 1 2 3
P(X=x) 0.343 0.441 0.189 0.027

 

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. What is the probability that none of the oil wells will be successful?

 

ANS:

P(X=0) = 0.343

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If a new pipeline will be constructed in the event that all three wells are successful, what is the probability that the pipeline will be constructed?

 

ANS:

P(X=3) = 0.027

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. How many of the wells can the company expect to be successful?

 

ANS:

0.9 wells

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. Suppose the first well to be completed is successful. What is the probability that one of the two remaining wells is successful?

 

ANS:

0.42

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

 

  1. If it costs $200,000 to drill each well and a successful well will produce $1,000,000 worth of oil over its lifetime, what is the expected net value of this three-well program?

 

ANS:

0.343(-$600,000)+0.441($400,000)+0.189($1,400,000)+0.027($2,400,000) = $300,000

 

PTS:   1                    MSC:  AACSB: Analytic | AACSB: Probability Concepts

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