Question

Chapter 10: Inferences About Means of Single Samples

7. Assume we know that σ ≥ 2, that the null hypothesis is Ho: µ = 7, and that the alternative hypothesis is H1: µ < 7. Consider the data Xi: 6, 3, 8, 4, 4, 2.

(a) What is the observed value of the statistic (Xobs)?

(b) What is the observed value of the test statistic (Zobs)?

(c) What is the critical value of the test statistic (Zcv) assuming the test is directional and α = .05?

(d) What is the critical value of the statistic (Xcv)?

(e) Should we reject the null hypothesis? Why?

Answer 1

Solution:-

a) The observed value of the statistics(Xobs) is 4.5.

b) The observed value of the test statistic (Zobs) is

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u > 7.0
Alternative hypothesis: u < 7.0

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample z-test.

Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).

SE = s / sqrt(n)

S.E = 0.8851

z = (x - u) / SE

z = 1.15

z_v = 1.645

Rejection region is z < -1.645.

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of 1.15.

Thus the P-value in this analysis is 0.151.

Interpret results. Since the P-value (0.151) is greater than the significance level (0.05), we have to reject the null hypothesis.

d)

z = (x - u) / SE

x = 5.544

e) Do not reject the null hypothesis.