MATH University of California Random Sample Size of a Normal Distribution Questions

Exercises 7.1-1, 7.1-4, 7.1-9, 7.2-1, 7.2-2, 7.2-5 in Probability and Statistical Inference (9th or10th edition).

7.1-1. A random sample of size 16 from the normal distri-bution N(?, 25) yielded x = 73.8. Find a 95% confidence interval for ?.

7.1-4. Let X equal the weight in grams of a “52-gram” snack pack of candies. Assume that the distribution of X is N(?, 4). A random sample of n = 10 observations of X yielded the following data:

55.95 56.06

56.54 59.93

57.58 58.30

55.13 52.57

57.48 58.46

(a) Give a point estimate for ?. (b) Find the endpoints for a 95% confidence interval for ?.

(c) On the basis of these very limited data, what is the probability that an individual snack pack selected at random is filled with less than 52 grams of candy?

7.1-9. During the Friday night shift, n = 28 mints were selected at random from a production line and weighed. They had an average weight of x = 21.45 grams and s = 0.31 grams. Give the lower endpoint of an approxi-mate 90% one-sided confidence interval for ?, the mean weight of all the mints.

7.2-1. The length of life of brand X light bulbs is assumed to be N(?X

is assumed to be N(?Y

, 784). The length of life of brand Y light bulbs , 627) and independent of X.Ifa

random sample of nX = 56 brand X light bulbs yielded = 57 brand Y light bulbs yielded a mean of y = 988.9

a mean of x = 937.4 hours and a random sample of size nY

hours, find a 90% confidence interval for ?X ? ?Y .

7.2-2. Let X1,X2,…,X5 be a random sample of SAT mathematics scores, assumed to be N(?X

,?2), and let

Y1,Y2,…,Y8 be an independent random sample of SAT verbal scores, assumed to be N(?Y

,?2). If the follow-ing data are observed, find a 90% confidence interval for ?X

? ?Y :

x1 = 644 x2 = 493 x3 = 532 x4 = 462 x5 = 565 y1 = 623 y2 = 472 y3 = 492 y4 = 661 y5 = 540 y6 = 502 y7 = 549 y8 = 518

7.2-5. A biologist who studies spiders was interested in comparing the lengths of female and male green lynx spiders. Assume that the length X of the male spider is approximately N(?X

,?2

observations of X: 5.20

4.70 5.70 5.65 5.75

4.70 4.80 6.20 5.50 5.95

Following are nY spider is approximately N(?Y,?2). Following are nX Y

5.75 5.95 5.40 5.65 5.90

7.50 5.20 6.20 5.85

6.45 6.35 5.85 5.75

7.00 6.10 = 30 observations of Y:

8.25 9.95 5.90

9.80 10.80 6.60 6.10 9.00 9.50

7.05 7.55

8.45 8.10

7.55 9.10

9.30 8.75 7.00 7.80 8.00 6.30 8.35 8.30 7.05

8.30 9.60

8.70 8.00 7.50 7.95

The units of measurement for both sets of observations are millimeters. Find an approximate one-sided 95% con-fidence interval that is an upper bound for ?X

? ?Y .