**1.175 Two Distributions** A market researcher selects a large sample of potential car buyers. For each consumer, she records gender, age, household income, and automobile preference. Which of these variables are categorical and which are quantitative? - Gender and automobile preference are categorical variables. - Age and household income are quantitative variables. If two distributions have exactly the same mean and standard deviation, must their histograms have the same shape? If they have the same five-number summary, must their histograms have the same shape? Explain. - Two distributions having the same mean and standard deviation do not necessarily have the same histogram shape. Histograms can differ in shape despite having identical means and standard deviations. - Distributions with the same five-number summary can also have different shapes, as the summary does not capture all aspects of the data's distribution. **1.176 Norms for Reading Scores** Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 70 and standard deviation 10 when given to third-graders. Sixth-graders have mean score 80 and standard deviation 11 on the same test. To provide separate “norms” for each grade, we transform the scores to have mean 100 and standard deviation 20. (a) What linear transformation will change third-grade scores \( x \) into new scores \( x_{\text{new}} = a + bx \) that have the desired mean and standard deviation? (Use \( b > 0 \) to preserve the order of the scores.) (b) Do the same for the sixth-grade scores. (c) David is a third-grade student who scores 72 on the test. Find David’s transformed score. Nancy is a sixth-grade student who scores 78. What is her transformed score? Who scores higher within his or her grade? (d) Suppose that the distribution of scores in each grade is Normal. Then both sets of transformed scores have the \( N(100, 20) \) distribution. What percent of third-graders have scores less than 75? What percent of sixth-graders have scores less than 75? **1.177 Use Software to Generate Some Data** Most statistical software packages have routines for generating values of variables having specified distributions. Use your statistical software to generate 30 observations from the \( N(25, 8) \) distribution. Compute the mean and standard deviation \( \bar{x} \) and \( s \) of the