Question

8. Consider a 90% confidence interval for µ not known. For which sample size, n = 10 or n = 20, is the confidence interval longer?
9. Critical Thinking Lorraine computed a confidence interval for µ based on
   a sample of size 41. Since she did not know α, she used s in her calculations.
   Lorraine used the normal distribution for the confidence interval instead of a
   Student's t distribution. Was her interval longer or shorter than one obtained
   by using an appropriate Student's t distribution? Explain.

11. Basic Computation: Confidence Interval Suppose x has a mound-shaped
    symmetric distribution. A random sample of size 16 has sample mean 10 and
    sample standard deviation 2.

1. Check Requirements Is it appropriate to use a Student's t distribution to
   compute a confidence interval for the population mean µ? Explain.

2. Find a 90% confidence interval for µ

(c) Interpretation Explain the meaning of the confidence interval you
computed.

In Problems 13 assume that the population of x values has an approximately
normal distribution.

13. Archaeology: Tree Rings: At Burnt Mesa Pueblo, the method of tree-ring
    dating gave the following years A.D. for an archaeological excavation site
    (Bandelier Archaeological Excavation Project: Summer 1990 Excavations at
    Burnt Mesa Pueblo, edited by Kohler, Washington State University):

                   1189 1271 1267 1272 1268 1316 1275 1317 1275

(a) Use a calculator with mean and standard deviation keys to verify that
the sample mean year is x = 1272, with sample standard deviation

s = 37years.

(b) Find a 90% confidence interval for the mean of all tree-ring dates from
this archaeological site.

(c) Interpretation What does the confidence interval mean in the context of
this problem?

Answer 1

Consider a 90% confidence interval for µ not known. For which sample size, n = 10 or n = 20, is the confidence interval longer?

For n = 10, the standard error of the mean will be larger and hence the confidence interval will be longer.

Degree of freedom = 41 - 1 = 40

Critical value of t for df = 40 and 95% confidence interval is 2.02

For normal distribution, critical value of z at 95% confidence interval is 1.96.

As the critical value of t is greater than that of z, the confidence interval would be longer for one obtained by using an appropriate Student's t distribution.

a.

The population distribution is moderately skewed, unimodal, without outliers because it is mound-shaped symmetric distribution, and the sample size is 16. So, it is appropriate to use a Student's t distribution to compute a confidence interval for the population mean.

b.

Degree of freedom = 16 - 1 = 15

Critical value of t for df = 15 and 90% confidence interval is 1.753

Standard error of mean = 2 / $\sqrt{16}$ = 0.5

90% confidence interval of the mean is,

(10 - 0.5 * 1.753, 10 + 0.5 * 1.753)

(9.1235, 10.8765)

c)

We are 90% confident that the population mean lies between 9.1235 and 10.8765.

13.

a)

Mean = 1272.222

Sample standard deviation = 36.92823

Thus, approximate value of mean and standard deviation is 1272 and 37 years.

(b)

Degree of freedom = 9 - 1 = 8

Critical value of t for df = 8 and 90% confidence interval is 1.86

Standard error of mean = 37 / $\sqrt{9}$ = 12.33

90% confidence interval of the mean is,

(1272 - 12.33 * 1.86, 1272 + 12.33 * 1.86)

(1249.066, 1294.934)

c)

We are 90% confident that the population mean of all tree-ring dates from this archaeological site lies between 1249.066 and 1294.934.