Question

2. According to a Gallup poll conducted in 2008, 78.3% of Americans felt satisfied with the way things were going in their personal lives. A researcher wonders of the percentage satisfied is lower today. To find out, the researcher surveys a random sample of 1,173 Americans, and finds that 895 feel satisfied with the way things are going in their personal lives. The researcher wishes to test the null hypothesis at a significance level of 0.05. (a) Conduct a large sample z test to determine whether there is sufficient evidence to reject the null hypothesis. Should you do a one-tailed test or a two-tailed test? (b) Interpret the meaning of the results in the context of this study.

Answer 1

Let p be the true proportion of Americans felt satisfied Today. A researcher wonders of the percentage satisfied is lower today. That is the researcher wonders if the percentage satisfied today is less than 0.783

The hypotheses are

a) We have the following information from the sample

The standard error of proportion is

The large sample test statistics is

This is a one tailed (left tailed) test (The alternative hypothesis has "<").

The critical value for alpha=0.05 is given by

Using standard normal table we get that for z=1.64 we get P(Z<1.64) = 0.5+0.4495=0.9495

Hence we will take the critical value as -1.64

We will reject the null hypothesis if the test statistics is less than the critical value -1.64.

Here, the test statistics is -1.66 and it is less than the critical value. Hence we reject the null hypothesis.

ans: there is sufficient evidence to reject the null hypothesis.

b) We can conclude that there is sufficient evidence to conclude that the percentage satisfied is lower today than in 2008 (lower than 78.3% satisfied, found in the Gallop poll conducted in 2008)