Question

# Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level...

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let α be 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 significance α and null hypothesis H0: μ = k, we reject H0 whenever k falls outside the c = 1 – α confidence interval for μ based on the sample data. When k falls within the c = 1 – α confidence interval, we do not reject H0.

(A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as p, μ1μ2, or p1p2, which we will study in later sections.) Whenever the value of k given in the null hypothesis falls outside the c = 1 – α confidence interval for the parameter, we reject H0. For example, consider a two-tailed hypothesis test with α = 0.10 and

H0: μ = 21
H1: μ ≠ 21

A random sample of size 17 has a sample mean x = 20 from a population with standard deviation σ = 8.

(a) What is the value of c = 1 − α?

Using the methods of Chapter 7, construct a 1 − α confidence interval for μ from the sample data. (Round your answers to two decimal places.)

 lower limit upper limit

What is the value of μ given in the null hypothesis (i.e., what is k)?
k =

Is this value in the confidence interval?

YesNo

Do we reject or fail to reject H0 based on this information?

Fail to reject, since μ = 21 is not contained in this interval.Fail to reject, since μ = 21 is contained in this interval.    Reject, since μ = 21 is not contained in this interval.Reject, since μ = 21 is contained in this interval.

(b) Using methods of Chapter 8, find the P-value for the hypothesis test. (Round your answer to four decimal places.)

Do we reject or fail to reject H0?

Reject the null hypothesis, there is sufficient evidence that μ differs from 21.Fail to reject the null hypothesis, there is insufficient evidence that μ differs from 21.    Fail to reject the null hypothesis, there is sufficient evidence that μ differs from 21.Reject the null hypothesis, there is insufficient evidence that μ differs from 21.

Compare your result to that of part (a).

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