Question

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, μ1 − μ2, or p1 − p2, which we will study later.) 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 = 0.01 and H0: μ = 20 H1: μ ≠ 20 A random sample of size 32 has a sample mean x = 23 from a population with standard deviation σ = 6. (a) What is the value of c = 1 − α? 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? Yes No Do we reject or fail to reject H0 based on this information? We fail to reject the null hypothesis since μ = 20 is not contained in this interval. We fail to reject the null hypothesis since μ = 20 is contained in this interval. We reject the null hypothesis since μ = 20 is not contained in this interval. We reject the null hypothesis since μ = 20 is contained in this interval. (b) Using methods of this chapter, find the P-value for the hypothesis test. (Round your answer to four decimal places.) Do we reject or fail to reject H0? We reject the null hypothesis since there is sufficient evidence that μ differs from 20. We fail to reject the null hypothesis since there is sufficient evidence that μ differs from 20. We fail to reject the null hypothesis since there is insufficient evidence that μ differs from 20. We reject the null hypothesis since there is insufficient evidence that μ differs from 20. Compare your result to that of part (a). We rejected the null hypothesis in part (a) but failed to reject the null hypothesis in part (b). We rejected the null hypothesis in part (b) but failed to reject the null hypothesis in part (a). These results are the same.

Answer 1

(a) What is the value of c = 1 − α?

Answer:- The value of c = 1 - 0.01 = 0.90

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 = 20

Is this value in the confidence interval?

No

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

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

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

The test statistic for the hypothesis test is :

                                             

So, the P-value is :-

Do we reject or fail to reject H0?

We reject the null hypothesis since there is sufficient evidence that μ differs from 20.

Compare your result to that of part (a).

These results are the same.