Question

A researcher claims the mean age of residents of a small town is more than 38 years. The age (in years) of a random sample of 30 sutdents are listed below. At alpha=0.10, is there enough evidence to support the researcher's claim? assum the population standard deviation is 9 years.

Ages (in years)
41
33
47
31
26
39
19
25
23
31
39
36
41
28
33
41
44
40
30
29
46
42
53
21
29
43
46
39
35
33
42
35
43
35
24
21
29
24
25
85
56
82
87
72
31
53
31
33
54
60
31
81
32
40
26
52
37
71

a) Identify the claim and state Ho and Ha
(b) Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed and whether to use a z-test, a t-test, or a chi-square test. Explain your reasoning
(c)Choose one of the options Option 1: Find the critical value(s), identify the rejection region(s), and find the appropriate standardized test statistic. Option 2: Find the appropriate standardized test statistic and the P-value
(d) Decide whether to reject or fail to reject the null hypothesis
(e) Interpret the decision in the context of the original claim.

Answer 1

a)

Hypotheses are:

(b)

Since population standard deviation is given and hypothesis is about population mean so single sample z test will be used.

(c)

Following is the output of descriptive statistics:

Descriptive statistics
	Ages (in years)
count	58
mean	40.60
sample standard deviation	16.40
sample variance	268.84
minimum	19
maximum	87
range	68

The sample mean is:

Option 1:

The test is right tailed so critical value of z for 0.10 level of significance is 1.28. The rejection region:

If z > 1.28, reject H0

The test statistics is:

Since test statistics lies in the rejection so we reject the null hypothesis.

Option 2:

The p-value using excel function "=1-NORMSDIST(2.2)" is:

p-value = P(z > 2.20) = 0.0139

Since p-value is less than level of significance so we reject the null hypothesis.

(d)

Reject the null hypothesis.

(e)

There is enough evidence to support the researcher claims that the mean age of residents of a small town is more than 38 years.