Is there a relationship between confidence intervals and two-tailed hypothesis tests? Letcbe the level of confidence used to construct a confidence interval from sample data. Letbe the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean.

For a two-tailed hypothesis test with level of significanceand null hypothesisH₀:=k,werejectH₀wheneverkfallsoutsidethec= 1confidence interval forbased on the sample data. Whenkfalls within thec= 1confidence interval, we do not rejectH₀.

(A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such asp,₁₂, orp₁p₂, which we will study later.) Whenever the value ofkgiven in the null hypothesis fallsoutsidethec= 1confidence interval for the parameter, wereject H₀. For example, consider a two-tailed hypothesis test with= 0.01and

H₀:= 20H₁:20

A random sample of size40has a sample meanx=23from a population with standard deviation=5.

(a) What is the value ofc= 1?

Construct a 1confidence interval forfrom the sample data. (Round your answers to two decimal places.)

lower limit=

upper limit=

What is the value ofgiven in the null hypothesis (i.e., what isk)?

k=

Is this value in the confidence interval?

Yes

No

Do we reject or fail to rejectH₀based on this information?

a. We fail to reject the null hypothesis since= 20 is not contained in this interval.

b. We fail to reject the null hypothesis since= 20 is contained in this interval.

c. We reject the null hypothesis since= 20 is not contained in this interval.

d. We reject the null hypothesis since= 20 is contained in this interval.

(b) Using methods of this chapter, find theP-value for the hypothesis test. (Round your answer to four decimal places.)

Do we reject or fail to rejectH₀?

a. We reject the null hypothesis since there is sufficient evidence thatdiffers from 20.

b. We fail to reject the null hypothesis since there is sufficient evidence thatdiffers from 20.

c. We fail to reject the null hypothesis since there is insufficient evidence thatdiffers from 20.

d. We reject the null hypothesis since there is insufficient evidence thatdiffers from 20.

Compare your result to that of part (a).

a. These results are the same.

b. We rejected the null hypothesis in part (a) but failed to reject the null hypothesis in part (b).

c. We rejected the null hypothesis in part (b) but failed to reject the null hypothesis in part (a).