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 μbased 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 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 = 0.05 and

H₀: μ = 21
H₁: μ ≠ 21

A random sample of size 14 has a sample mean x = 19 from a population with standard deviation σ = 5.

(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?

Yes

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

Fail to 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?

Fail to 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.

Answer 1