Question

3.26Daily activity. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking.¹⁰ Among mildly obese people, minutes of activity varied according to the N(373, 67) distribution. Minutes of activity for lean people had the N(526, 107) distribution. Within what limits do the active minutes for about 95% of the people in each group fall? Use the 68–95–99.7 rule.

3.27Low IQ test scores. Scores on the Wechsler Adult Intelligence Scale (WAIS) are approximately Normal with mean 100 and standard deviation 15. People with WAIS scores below 70 are considered intellectually disabled when, for example, applying for Social Security disability benefits. According to the 68–95–99.7 rule, about what percent of adults are intellectually disabled by this criterion?

3.28Standard Normal drill.In each case, sketch a standard Normal curve and shade the area representing the region.

(a)z ≤ −2.15

(b)z ≥ −2.15

(c)z > 1.57

(d)−2.15 < z < 1.57

3.29Standard Normal drill.

(a)Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.3.

(b)Find the number z such that 35% of all observations from a standard Normal distribution are greater than z.

3.30Fruit flies. The common fruit fly Drosophila melano-gaster 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 millimeters (mm) and standard deviation 0.078 mm.

(a)What proportion of flies have thorax length less than 0.6 mm?

(b)What proportion have thorax length greater than 0.9 mm?

(c)What proportion have thorax length between 0.6 mm and 0.9 mm?

3.33A milling machine. Automated manufacturing operations are quite precise but still vary, often with distributions that are close to Normal. The width in inches of slots cut by a milling machine follows approximately the N(0.8750, 0.0012) distribution. The specifications allow slot widths between 0.8725 and 0.8775 inch. What proportion of slots meet these specifications?

3.37The middle half. The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. They span the middle half of the distribution. What are the quartiles of the distribution of gas mileage?

3.44Grading managers. Some companies “grade on a bell curve” to compare the performance of their managers and professional workers. This forces the use of some low performance ratings so that not all workers are listed as “above average.” Ford Motor Company’s “performance management process” for this year assigned 10% A grades, 80% B grades, and 10% C grades to the company’s managers. Suppose Ford’s performance scores really are Normally distributed. This year, managers with scores less than 25 received C grades and those with scores above 475 received A grades. What are the mean and standard deviation of the scores?

Answer 1

Given,3.26

Daily activity of obese people and lean people follow normal distribution

As we know, in a normal distribution

68% values fall under mean±1SD

95% values fall under mean±2SD

99.7% values fall under mean±3SD

to find the lower and upper limits for 95% of the people in both obese and lean group

we have,

mean of obese=373

SD of obese=67

upper limit=mean+2SD

upper limit =373+2*67

upper limit =507

Lower limit=mean-2SD

Lower limit =373-2*67

Lower limit =239

hence the limits of active minutes for 95% of obese people fall under (239,507)

mean of lean=373

SD of lean=67

upper limit=mean+2SD

upper limit =526+2*107

upper limit =640

Lower limit=mean-2SD

Lower limit =526-2*107

Lower limit =312

hence the limits of active minutes for 95% of lean people fall under (312,640)

Question 3.27:

Ans

According to empirical rule of normal distribution, 95% of the observations lies within 2 standard deviations from the mean. Therefore (100 - 95)/2 = 2.5% lies outside 2 standard deviations on both the sides.

70 = 100 - 2*15 that is 2 standard deviations less than the mean.

Therefore according to the given empirical rule, about 2.5% of the adults are retarded by the given criterion.