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Monsoon Rains. The summer monsoon rains bring \(80 \%\) of India's rainfall and are essential for the country's agriculture. Records going back more than a century show that the amount of monsoon rainfall varies from year to year according to a distribution that is approximately Normal with mean 852 millimeters \((\mathrm{mm})\) and standard deviation \(82 \mathrm{~mm}^{3}\) Use the \(68-95-99.7\) rule to answer the following questions. (a) Between what values do the monsoon rains fall in the middle \(95 \%\) of all years? (b) How small are the monsoon rains in the driest \(2.5 \%\) of all years?

Men's and Women's Heights. The heights of women aged \(20-29\) in the United States are approximately Normal with mean \(64.2\) inches and standard deviation \(2.8\) inches. Men the same age have mean height \(69.4\) inches with standard deviation \(3.0\) inches. \({ }^{6}\) What are the \(z\)-scores for a woman \(5.5\) feet tall and a man \(5.5\) feet tall? Say in simple language what information the \(z\)-scores give that the original nonstandardized heights do not.

Use the Normal Table. Use Table A to find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question. (a) \(z<-0.42\) (b) \(z>-1.58\) (c) \(z<2.12\) (d) \(-0.42

The Medical College Admissions Test. Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). \({ }^{8}\) A new version of the exam was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded has been modified, with the total score of the four sections on the test ranging from 472 to 528 . In spring 2015 , the mean score was \(500.0\) with a standard deviation of \(10.6\). (a) What proportion of students taking the MCAT had a score over 510 ? (b) What proportion had scores between 505 and 515 ?

The Medical College Admissions Test. A new version of the Medical College Admissions Test (MCAT) was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded has been modified, with the total score of the four sections on the test ranging from 472 to 528 . In spring 2015 , the mean score was \(500.0\) with a standard deviation of \(10.6\) (a) What are the median and the first and third quartiles of the MCAT scores? What is the interquartile range? (b) Give the interval that contains the central \(80 \%\) of the MCAT scores.

Fruit flies. The common fruit fly Drosophila melanogaster is the most studied organism in genetic research because it is small, is easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit flies is approximately Normal with mean \(0.800\) millimeter (mm) and standard deviation \(0.078 \mathrm{~mm}\). (a) What proportion of flies have thorax length less than \(0.7 \mathrm{~mm}\) ? (b) What proportion have thorax length greater than \(1.0 \mathrm{~mm}\) ? (c) What proportion have thorax length between \(0.7 \mathrm{~mm}\) and \(1.0 \mathrm{~mm}\) ?

Acid rain? Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by \(\mathrm{pH}\) on a scale of 0 to 14 . Distilled water has \(\mathrm{pH} 7.0\), and lower \(\mathrm{pH}\) values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below \(5.0\). The \(\mathrm{pH}\) of rain at one location varies among rainy days according to a Normal distribution with mean \(5.43\) and standard deviation \(0.54\). What proportion of rainy days have rainfall with \(\mathrm{pH}\) below \(5.0\) ?

Are we getting smarter? When the Stanford-Binet IQ test came into use in 1932 , it was adjusted so that scores for each age group of children followed roughly the Normal distribution with mean 100 and standard deviation 15 . The test is readjusted from time to time to keep the mean at 100 . If present-day American children took the 1932 Stanford-Binet test, their mean score would be about 120 . The reasons for the increase in IQ over time are not known but probably include better childhood nutrition and more experience in taking tests. 11 (a) IQ scores above 130 are often called "very superior." What percentage of children had very superior scores in 1932 ? (b) If present-day children took the 1932 test, what percentage would have very superior scores? (Assume that the standard deviation 15 does not change.)

Perfect SAT scores. It is possible to score higher than 1600 on the combined mathematics and reading portions of the SAT, but scores 1600 and above are reported as 1600 . The distribution of SAT scores (combining mathematics and reading) in 2014 was close to Normal with mean 1010 and standard deviation 218. What proportion of SAT scores for these two parts were reported as 1600 ? (That is, what proportion of SAT scores were actually higher than 1600?)