91Ó°ÊÓ

Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The area below 40 for a \(N(48,5)\) distribution converted to a standard normal distribution

Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The upper \(30 \%\) for a \(N(48,5)\) distribution converted to a standard normal distribution

The Scholastic Aptitude Test (SAT) was taken by 1,698,521 college-bound students in the class of \(2015 .^{24}\) The test has three parts: Critical Reading, Mathematics, and Writing. Scores on all three parts range from 200 to 800 . The means and standard deviations for the three tests are shown in Table P.16. Assuming that the Critical Reading scores follow a normal distribution, draw a sketch of the normal distribution and label at least 3 points on the horizontal axis. $$ \begin{array}{lcc} & \text { Mean } & \text { St. Dev. } \\ \hline \text { Critical Reading } & 495 & 116 \\ \text { Mathematics } & 511 & 120 \\ \text { Writing } & 484 & 115 \end{array} $$

Critical Reading on the SAT Exam In Table P.16 with Exercise \(\mathrm{P} .157\), we see that scores on the Critical Reading portion of the SAT (Scholastic Aptitude Test) exam are normally distributed with mean 495 and standard deviation \(116 .\) Use the normal distribution to answer the following questions: (a) What is the estimated percentile for a student who scores 700 on Critical Reading? (b) What is the approximate score for a student who is at the 30 th percentile for Critical Reading?

Heights of Men in the US Heights of adult males in the US are approximately normally distributed with mean 70 inches \((5 \mathrm{ft} 10 \mathrm{in})\) and standard deviation 3 inches. (a) What proportion of US men are between \(5 \mathrm{ft}\) 8 in and \(6 \mathrm{ft}\) tall \((68\) and 72 inches, respectively)? (b) If a man is at the 10 th percentile in height, how tall is he?

Exam Grades Exam grades across all sections of introductory statistics at a large university are approximately normally distributed with a mean of 72 and a standard deviation of 11 . Use the normal distribution to answer the following questions. (a) What percent of students scored above a \(90 ?\) P.167 Exam Grades Exam grades across all sections of introductory statistics at a large university are approximately normally distributed with a mean of 72 and a standard deviation of \(11 .\) Use the normal distribution to answer the following questions. (a) What percent of students scored above a \(90 ?\)

Curving Grades on an Exam A statistics instructor designed an exam so that the grades would be roughly normally distributed with mean \(\mu=75\) and standard deviation \(\sigma=10 .\) Unfortunately, a fire alarm with ten minutes to go in the exam made it difficult for some students to finish. When the instructor graded the exams, he found they were roughly normally distributed, but the mean grade was 62 and the standard deviation was 18\. To be fair, he decides to "curve" the scores to match the desired \(N(75,10)\) distribution. To do this, he standardizes the actual scores to \(z\) -scores using the \(N(62,18)\) distribution and then "unstandardizes" those \(z\) -scores to shift to \(N(75,10)\). What is the new grade assigned for a student whose original score was 47 ? How about a student who originally scores a \(90 ?\)

Empirical Rule for Normal Distributions Pick any positive values for the mean and the standard deviation of a normal distribution. Use your selection of a normal distribution to answer the questions below. The results of parts (a) to (c) form what is often called the Empirical Rule for the standard deviation in a normal distribution. (a) Verify that about \(95 \%\) of the values fall within two standard deviations of the mean. (b) What proportion of values fall within one standard deviation of the mean? (c) What proportion of values fall within three standard deviations of the mean? (d) Will the answers to (b) and (c) be the same for any normal distribution? Explain why or why not.