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 30 has a sample mean x = 23 from a population with standard deviation σ = 6.

(a) What is the value of c = 1 − α? 2.826 Incorrect: Your answer is incorrect.

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 Correct: Your answer is correct. 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. Correct: Your answer is correct.

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

Answer 1

a) The value of c is = 1 − α = 1 - 0.01 = 0.99

b)

a 99% confidence interval for μ from the sample data = (20.18, 25.82)

Lower limit = 20.18

Upper limit = 25.82

k = 20, here 20 is outside the confidence interval.

ans-> No

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

b)