91Ó°ÊÓ

Mensa is an organization designed for people of high intelligence. One qualifies for Mensa if one's intelligence is measured at or above the 98 th percentile. Explain what this means.

Explain the advantage of using \(z\) -scores to compare observations from two different data sets.

Use the five test scores of 65,70,71 \(75,\) and 95 to answer the following questions: (a) Find the sample mean. (b) Find the median. (c) Which measure of central tendency better describes the typical test score? (d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores. (e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Compute the sample standard deviation of the following test scores: 78,78,78,78 . What can be said about a data set in which all the values are identical?

The Fair Isaacs Corporation has devised a model that is used to determine creditworthiness of individuals, called a FICO score. FICO scores range in value from 300 to 850 , with a higher score indicating a more creditworthy individual. The distribution of FICO scores is skewed left with a median score of \(723 .\) (a) Do you think the mean FICO score is greater than, less than, or equal to 723 ? Justify your response. (b) What proportion of individuals have a FICO score above \(723 ?\)

The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. $$ \begin{array}{lll} \hline 2.0 & 1.9 & 1.8 \\ \hline 3.2 & 2.4 & 3.4 \\ \hline 1.6 & 2.7 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 8.4 & 7.2 & 7.6 \\ \hline 7.4 & 6.9 & 9.4 \\ \hline 9.1 & 8.1 & \\ \hline \end{array} $$ (a) Determine the mean and standard deviation of each data set. (b) Based only on the standard deviation, which data set has more spread? (c) What proportion of the observations is within one standard deviation of the mean for each data set? (d) The coefficient of variation, \(C V\), is defined as the ratio of the standard deviation to the mean of a data set, so $$ C V=\frac{\text { standard deviation }}{\text { mean }} $$ The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more "spread"? (e) Let's take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters. $$ \begin{array}{lll} \hline 74 & 68 & 71 \\ \hline 66 & 72 & 69 \\ \hline 69 & 71 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 187.96 & 172.72 & 180.34 \\ \hline 167.64 & 182.88 & 175.26 \\ \hline 175.26 & 180.34 & \\ \hline \end{array} $$

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning. (a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011 (b) Most popular major for students enrolled in a statistics course (c) Average test score when the scores are distributed symmetrically (d) Average test score when the scores are skewed right (e) Average income of a player in the National Football League (f) Most requested song at a radio station (g) Typical number on the jersey of a player in the National Hockey League.

What makes the range less desirable than the standard deviation as a measure of dispersion?

In one of Sullivan's statistics sections, the standard deviation of the heights of all students was 3.9 inches. The standard deviation of the heights of males was 3.4 inches and the standard deviation of females was 3.3 inches. Why is the standard deviation of the entire class more than the standard deviation of the males and females considered separately?