Chapter 3 - Managerial Statistics

A doctor records a pair of information (temperature when admitted, temperature 24 hours later) for 400 hospital patients selected at random. He finds that the average patients' temperature upon admission is 101 degrees and 99 degrees 24 hours later. The standard deviation is 1.5 degrees for temperatures when admitted and .5 degrees 24 hours later. The correlation between the two temperatures is .8. What is the covariance between the variables?

.60 The covariance between the variables is .60. See Section 3.5, Measures of Association Between Two Variables.

The correlation coefficient between two scores X and Y equals .80. If both the X scores and the Y scores are converted to z-scores, then the correlation between the z-scores for X and the z-scores for Y would be:

.80 The correlation between the z-scores for X and the z-scores for Y would be .80. See Section 3.5, Measures of Association Between Two Variables.

(see image 7) Use Exhibit 3.3a for question 2. Assuming a bell-shaped distribution with mean of 66 and standard deviation of 2, calculate the z-score for X = 68 and X = 69, respectively.

1 and 1.5 The z-scores for the two X values are 1 and 1.5, respectively. See Section 3.3, Measures of Distribution Shape, Relative Location, and Detecting Outliers.

The above is a histogram showing the actual frequency of an average college professor's salary from 50 randomly selected colleges. Based on the frequency histogram for salaries, the bin that contains the 80th percentile is: (see image 2)

63-68 The value 65 is at the 80th percentile. The location of the pth percentile can be calculated using the formula . See Section 3.1, Measures of Location.

In a sample of 500 students whose mean height is 67.8 inches, 150 were women. If the mean height of the women was 63.0 inches, what is the mean height of the men in the sample?

69.9 The mean height of men in the sample is 69.9 inches; that is See Section 3.1, Measures of Location.

A student had scores of 85, 56, and 91 on her first three statistics tests. What score does she need to get on her next test to have a test average of 80?

88 She needs to score an 88. The score can be calculated using the equation (see equation after image 3). See Section 3.1, Measures of Location.

An outlier should remain in the data set under what circumstances?

An outlier should remain in the data set when an unusual data value has been recorded correctly. If an unusual data value has been recorded correctly, it should remain in the data set. See Section 3.3, Measures of Distribution Shape, Relative Location, and Detecting Outliers.

Which of the following would not be a correct interpretation of correlation coefficient of r = -.89?

Eighty-nine percent of the variation between the two variables in the linear regression is explained. The statement "89% of the variation between the two variables in the linear regression is explained" is incorrect. See Section 3.5, Measures of Association Between Two Variables.

Which of the following is not resistant to the outliers in a data set?

Mean The mean is nonresistant and is affected by outliers in a data set. See Section 3.1, Measures of Location.

If the variance of a data set is correctly computed with the formula using in the denominator, which of the following is true?

The data set is a sample. The data set is a sample if the variance formula is using in the denominator. See Section 3.2, Measures of Variability.

Growth factors for the population of Dallas in the past five years have been 1, 2, 3, 4, and 5, respectively. The geometric mean is:

The geometric mean is the nth root of the product of n factors. The product of 1, 2, 3, 4, and 5 is 120. The fifth root of 120 is (120)⅕. See Section 3.1, Measures of Location.

In a five-number summary, which of the following is not used for data summarization?

The mean The mean is not used for data summarization in the five-number summary. See Section 3.4, Five-Number Summaries and Boxplots.

A student was performing an experiment that compared a new high protein food to the old food for pigs. He found the mean weight gain for subjects consuming the new food to be 12.8 pounds with a standard deviation of 3.5 pounds. Later, he realized that the scale was out of calibration by 1.5 pounds (meaning that the scale weighed items 1.5 pounds more). What should the mean and standard deviation be for the subjects consuming the new food?

The mean should be 11.3, and the standard deviation should remain unchanged. The mean should be 11.3 (12.8 minus the overage), and the standard deviation should remain unaffected. If the population of pigs were systematically overweighed, then their average deviation from the mean should not change. See Section 3.1, Measures of Location.

When testing water for chemical impurities, results are often reported as BDL, that is, below detection limit. The following data set gives the measurements of the amount of lead in a series of water samples taken from inner-city households (ppm): 5, 7, 12, BDL, 10, 8, BDL, 20, 6 Which of the following is correct?

The median lead level in the water is 7 ppm. First, order the values from least to greatest: BDL, BDL, 5, 6, 7, 8, 10, 12, and 20. The middle value, 7, is the median. See Section 3.1, Measures of Location.

Look at this boxplot. What can you tell about the data set it comes from? (see image 8)

This data set is approximately symmetric. This data set is approximately symmetric. See Section 3.4, Five-Number Summaries and Boxplots.

The measure of location that is most likely to be influenced by extreme values in a data set is the:

mean The measure of location that is most likely to be influenced by extreme values in a data set is the mean. The mean is a non-resistant statistic. It will be pulled in the direction of the outlier. See Section 3.1, Measures of Location.

The interquartile range is:

the difference between Q3 and Q1, the third and first quartiles. The interquartile range is the difference between Q3 and Q1, the third and first quartiles. See Section 3.2, Measures of Variability.

There are three executives in an office with ages of 56, 57, and 58. If a 57-year-old executive enters the room, then:

the mean age will stay the same, but the variance will decrease. The mean age will stay the same, but the variance will decrease. See Section 3.2, Measures of Variability.

Which one of the following statistics measures both the strength and direction of a linear relationship?

Correlation coefficient The variable rxy, the correlation coefficient, measures the strength and direction of a linear relationship. See Section 3.5, Measures of Association Between Two Variables.

Exhibit for question 5HISTOGRAM OF TOP 10 COUNTRIES FOR PER CAPITA GDP 2014 (see image 3) What is the 75th percentile of the top ten countries as ranked for per capita GDP in 2014?

$89,400 The 75th percentile is the upper quartile of the data set (Q3). This calculation is obtained from the five number summary. The data is already organized in ascending order. The upper quartile is the median of the upper half of the data. It can also be identified from a percentile plot or a boxplot. See Section 3.4, Five-Number Summaries and Boxplots.

In a sample of size five, the mean is 23 and four of the observations have the following deviations from the mean: -6, 2, 5, and 3. What is the value of the fifth observation?

-4 The value of the fifth observation is -4. The sum of deviations from the mean must sum to 0. See Section 3.2, Measures of Variability.

Which correlation coefficient best matches the data shown in Scatter Plot 1?

-.5 The scatter plot indicates a moderate positive relationship. A correlation coefficient of .6 is the only option that indicates a moderate positive relationship. See Section 3.5, Measures of Association Between Two Variables.

The correlation coefficient ranges between:

-1 and 1 The correlation coefficient ranges between -1 and 1 inclusive. See Section 3.5, Measures of Association Between Two Variables.

Suppose we have the following data: 12, 17, 13, 25, 16, 21, 30, 14, 16, and 18. To find the 10% trimmed mean, what numbers should be deleted from the calculation?

12 and 30 The 10% trimmed mean would eliminate the lowest and highest values in this data set (12 and 30). See Section 3.1, Measures of Location.

Given the following information: Standard deviation = 8Coefficient of variation = 64%The mean is

12.5 The mean can be calculated using (see formula after image 4) . See Section 3.2, Measures of Variability.

The standard deviation of a sample was reported to be 20.The report indicated that = 7200. What is the sample size?

19 The sample size would be 19. The sample size can be found by solving for n in the sample variance formula: See Section 3.2, Measures of Variability.

The summary statistics for the hourly wages of a sample of 130 system analysts are given below: Mean = 60Range = 20Mode = 73Variance = 324Median = 74 The coefficient of variation is equal to:

30% The coefficient of variation is (see formula after image 5). See Section 3.2, Measures of Variability.

A data set has the following five-number summary: {31, 50, 58, 62, and 87}. Which of the following pairs of values in this data set would be considered outliers?

31 and 81 An outlier is any value that is more than 1.5(IQR) above Q3 or below Q1. The values 31 and 81 would be considered outliers. See Section 3.4, Five-Number Summaries and Boxplots.

The heights of adult women are approximately normally distributed about a mean of 65 inches, with a standard deviation of 2 inches. If Rachel is at the 95th percentile in height for adult women, then her height is closest to:

69 inches Using the Empirical rule, approximately 68% of the observations are within 1 standard deviation of the mean. Approximately 95% are within 2 standard deviations of the mean. By drawing a normal curve, it can be seen that 69 is the closest to the 95th percentile. See Section 3.3, Measures of Distribution Shape, Relative Location, and Detecting Outliers.

A survey conducted a local university asked students how many hours they studied a week. The survey showed that on average, students at this university study 7.5 hours per week with a standard deviation of 1.25 hours. What percentage of students study between 5 hours and 10 hours per week?

75% The data values of 5 and 10 hours per week are 2 standard deviations away from the mean. According to Chebyshev's theorem, 75% of the data values must be within 2 standard deviations from the mean. See Section 3.3, Measures of Distribution Shape, Relative Location, and Detecting Outliers.

Which of the above graphs displays the strongest positive correlation coefficient? (see image 1 Ch. 3 word )

Graph 1 Graph 1 displays the strongest positive correlation coefficient. See Section 3.5, Measures of Association Between Two Variables.

Which of the above graphs displays no linear relationship between the x and y variables? (see image 5)

Graph 2 Graph 2 displays no linear relationship between the x and y variables. See Section 3.5, Measures of Association Between Two Variables.

Which of the above graphs displays the weakest linear relationship between the x and y variables? (see image 6)

Graph 3 Graph 3 displays the weakest linear relationship between the x and y variables. See Section 3.5, Measures of Association Between Two Variables.

Which of the above graphs displays the strongest linear relationship between the x and y variables? (see image 9)

Graph 4 Graph 4 displays the strongest linear relationship between the x and y variables. See Section 3.5, Measures of Association Between Two Variables.

Consider these parallel boxplots of gasoline mileage for three makes of cars. Which of the following are true statements? I. All three have the same range.II. All three have the same interquartile range.III. The difference in the medians between the first and third distributions is equal to the interquartile range of the second distribution. (see image 4)

I and III All three have the same range and the difference in the medians between the first and third distributions is equal to the interquartile range of the second distribution. See Section 3.4, Five-Number Summaries and Boxplots.

Which of the following provides a measure of central location for the data?

Mean The mean provides a measure of central location for the data. It can be viewed as the fulcrum or balancing point of a distribution. See Section 3.1, Measures of Location.

When a set of data has suspect outliers, which of the following is the referred measure of central tendency?

Median If a data set has suspect outliers, the measure of central tendency that is resistant to extreme values, i.e. the median, should be used. This would be the median. See Section 3.1, Measures of Location.

When data are positively skewed, the mean will usually be:

greater than the median. When data are positively skewed, the mean will usually be greater than the median. See Section 3.3, Measures of Distribution Shape, Relative Location, and Detecting Outliers.

The correlation coefficient is:

a number between -1 and 1 inclusive that measures the strength and direction of the linear relationship between two numerical variables. The correlation coefficient is a number between -1 and 1 inclusive that measures the strength and direction of the linear relationship between two numerical variables. See Section 3.5, Measures of Association Between Two Variables.

Positive values of covariance indicate:

a positive relation between the x values and y values. Positive values of covariance indicate a positive relation between the independent and the dependent variables (x and y values). Covariance implies direction of a bivariate relationship. See Section 3.5, Measures of Association Between Two Variables.

The sum of deviations of the individual data elements from their mean is:

always equal to zero. The sum of deviations of the individual data elements from their mean is always equal to zero. See Section 3.2, Measures of Variability.

A numerical measure of linear association between two variables is the:

covariance Covariance implies direction of a bivariate relationship. See Section 3.5, Measures of Association Between Two Variables.

A recent study found that hamburger fat calories "x" had a positive linear association with the amount of sodium "y" found in that hamburger. This can be interpreted to indicate that:

hamburgers with a low amount of fat calories tend to have a low amount of sodium. Low fat would correlate with low sodium. See Section 3.5, Measures of Association Between Two Variables.

The pth percentile is a value such that at least p percent of the observations are:

less than or equal to this value. The pth percentile is a value such that at least p percent of the observations are less than or equal to this value. Percentile rank assumes cumulative relative frequency. All the values below a certain observation are added together with the pth value. See Section 3.1, Measures of Location. Additional Resources

The numerical value of the standard deviation can never be:

negative The numerical value of the standard deviation can never be negative. See Section 3.2, Measures of Variability.

The difference between the largest and the smallest data values is the:

range The difference between the largest and the smallest data values is the range. See Section 3.2, Measures of Variability.

The interquartile range is used as a measure of variability to overcome the fact that the:

range is comprised of extreme values. The interquartile range is used as a measure of variability to overcome the fact that the range is comprised of extreme values. See Section 3.2, Measures of Variability.

A numerical value used as a summary measure for a sample, such as sample mean, is known as a:

sample statistic. A numerical value used as a summary measure for a sample, such as sample mean, is known as a sample statistic. Summary measures are taken for both samples and populations. Samples are described by statistics. Populations are described by parameters. See Section 3.1, Measures of Location.

A sample of 99 distances has a mean of 24 feet and a median of 21.5 feet. Unfortunately, it has just been discovered that an observation that was erroneously recorded as "30" actually had a value of "35." If we make this correction to the data, then:

the median remains the same, but the mean is increased. In a data set with 99 values, the median would remain the same, but the mean would increase. See Section 3.1, Measures of Location.

The coefficient of variation is:

the standard deviation divided by the mean and multiplied by 100. The coefficient of variation is the standard deviation divided by the mean and multiplied by 100. See Section 3.2, Measures of Variability.

Suppose the correlation coefficient rxy between the amount of sleep (in hours) "x" and number of yawns made in 8:00 a.m. classes "y" of 100 business statistics students is computed to be -.82. Then:

there is a strong negative linear relationship between the two variables. A correlation coefficient of -.82 implies a linear relationship that is strong and negative. See Section 3.5, Measures of Association Between Two Variables.