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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
 A sample of a population taken at one particular point in time is categorized as:
a.  categorical  c.  crosssectional 
b.  discrete  d.  timeseries 
ANS: C PTS: 1 MSC: AACSB: Analytic  AACSB: Statistical Inference
 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
 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
 Numerical variables can be subdivided into which two types?
a.  Diverse and categorical  c.  Nominal and progressive 
b.  Discrete and continuous  d.  Crosssectional and discrete 
ANS: B PTS: 1 MSC: AACSB: Analytic
 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
 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
 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
 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
 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
 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
 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
 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
 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
 Which of the following are possible categorizations of data type?
a.  Numerical versus categorical (with subcategories nominal, ordinal) 
b.  Discrete versus continuous 
c.  Crosssectional versus time series 
d.  All of these options 
e.  Two of these options 
ANS: D PTS: 1 MSC: AACSB: Analytic
 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
 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 50^{th} percentile 
d.  All of these options 
ANS: D PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 If a value represents the 95^{th} 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
 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
 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
 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
 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
 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
 The mode is best described as the
a.  middle observation 
b.  same as the average 
c.  50^{th} percentile 
d.  most frequently occurring value 
e.  third quartile 
ANS: D PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 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
 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
 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
 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
 Expressed in percentiles, the interquartile range is the difference between the
a.  10^{th} and 60^{th} percentiles 
b.  15^{th} and 65^{th} percentiles 
c.  20^{th} and 70^{th} percentiles 
d.  25^{th} and 75^{th} percentiles 
e.  35^{th} and 85^{th} percentiles 
ANS: D PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 Age, height, and weight are examples of numerical data.
ANS: T PTS: 1 MSC: AACSB: Analytic
 Data can be categorized as crosssectional or time series.
ANS: T PTS: 1 MSC: AACSB: Analytic
 All nominal data may be treated as ordinal data.
ANS: F PTS: 1 MSC: AACSB: Analytic
 Four different shapes of histograms are commonly observed: symmetric, positively skewed, negatively skewed, and bimodal.
ANS: T PTS: 1 MSC: AACSB: Analytic
 Categorical variables can be classified as either discrete or continuous.
ANS: F PTS: 1 MSC: AACSB: Analytic
 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
 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
 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
 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
 In the term “frequency table,” frequency refers to the number of data values falling within each category.
ANS: T PTS: 1 MSC: AACSB: Analytic
 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
 The number of car insurance policy holders is an example of a discrete random variable
ANS: T PTS: 1 MSC: AACSB: Analytic
 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
 Phone numbers, Social Security numbers, and zip codes are examples of numerical variables.
ANS: F PTS: 1 MSC: AACSB: Analytic
 Crosssectional 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
 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
 Both ordinal and nominal variables are categorical.
ANS: T PTS: 1 MSC: AACSB: Analytic
 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
 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
 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
 The mean is a measure of central location.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 The length of the box in the boxplot portrays the interquartile range.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 The value of the standard deviation always exceeds that of the variance.
ANS: F PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 The difference between the first and third quartiles is called the interquartile range.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 The standard deviation is measured in original units, such as dollars and pounds.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 The median is one of the most frequently used measures of variability.
ANS: F PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 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
 Expressed in percentiles, the interquartile range is the difference between the 25^{th} and 75^{th} percentiles.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 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
 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
 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
 The median of a data set with 30 values would be the average of the 15^{th} and the 16^{th} 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
 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 25^{th} percentile for men, you would be above the 50^{th} 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
 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
 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
 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
 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
 Find the mean, median, standard deviation, first and third quartiles, and the 95^{th} percentile for family incomes in both years.
ANS:
Income 1980 Income 1990
Mean
Median Standard deviation First quartile Third quartile 95^{th} 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
 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 95^{th} percentile shows that the mean increase might be because the rich got richer.
PTS: 1 MSC: AACSB: Analytic  AACSB: Statistical Inference
 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 fastfood restaurant would like to know the dispersion of customer service times about their average value for the facility’s driveup 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
 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
 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
 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
 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
 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
 What salary amount represents the second quartile?
ANS:
$35,000 (median)
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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 50^{th} 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
 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
 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
 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
 Compute the mean number of children.
ANS:
Mean = 1.90
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 Compute the median number of children.
ANS:
Median = 1.5
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 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
 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
 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 postsecondary education, and annual salary.
NARREND
 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
 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 3040 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
 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
 What percentage of the job applicants scored between 30 and 40?
ANS:
10%
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 What percentage of the job applicants scored below 60?
ANS:
90%
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 How many job applicants scored between 10 and 30?
ANS:
100
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 How many job applicants scored above 50?
ANS:
50
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 Seventy percent of the job applicants scored above what value?
ANS:
20
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 Half of the job applicants scored below what value?
ANS:
30
PTS: 1 MSC: AACSB: Analytic  AACSB: Descriptive Statistics
 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
 Probabilities that cannot be estimated from longrun 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
 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
 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
 Probabilities that can be estimated from longrun 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
 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
 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
 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
 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
 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
 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 longrun proportions 
d.  is the distribution of a single random variable 
ANS: A PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 If P(A) = P(AB), 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 If A and B are any two events with P(A) = .8 and P(BA) = .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
 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
 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
 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
 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
 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
 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
 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
 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
 The probability that event A will not occur is denoted as .
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 If P(A and B) = 1, then A and B must be collectively exhaustive.
ANS: T PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 When two events are independent, they are also mutually exclusive.
ANS: F PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 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
 The multiplication rule for two events A and B is: P(A and B) = P(AB)P(A).
ANS: F PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 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
 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
 If P(A and B) = 0, then A and B must be collectively exhaustive.
ANS: F PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 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
 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
 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
 (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
 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
 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
 (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
 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
 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
 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
 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
 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
 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
 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
 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
 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 > 2500P<30)=(.03 + .15 + .10 + .05)/(.28 + .26) = .33/.54 = .6111
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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 < 3500P > 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
 Areand_{ }independent random variables? Explain why or why not.
ANS:
No. The, this means that given that has occurred, this changes the probability of X_{1 }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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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+280208)/1000 = 552/1000 = 0.552
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 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
 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
 What is the probability a randomly selected patron prefers wine?
ANS:
P(W) = 0.4
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 What is the probability a randomly selected patron is a female?
ANS:
P(F) = 0.30
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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
 Suppose a randomly selected patron prefers wine. What is the probability the patron is a male?
ANS:
P(MW) = 0.35
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 Suppose a randomly selected patron prefers beer. What is the probability the patron is a male?
ANS:
P(MB) = 0.933
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 Suppose a randomly selected patron is a female. What is the probability the patron prefers beer?
ANS:
P(BF) = 0.133
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 Suppose a randomly selected patron is a female. What is the probability that the patron prefers wine?
ANS:
P(WF) = 0.867
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 Are gender of patrons and drinking preference independent? Explain.
ANS:
P(WF) = 0.867, and P(W) = 0.40. Since P(WF) 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
 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
 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
 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
 How many of the wells can the company expect to be successful?
ANS:
0.9 wells
PTS: 1 MSC: AACSB: Analytic  AACSB: Probability Concepts
 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
 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 threewell 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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