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 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 = 0.01 and

H₀: μ = 21
H₁: μ ≠ 21

A random sample of size 22 has a sample mean x = 19 from a population with standard deviation σ = 6.

(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 No    

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).

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

Answer 1

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 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 = 0.01 and

H₀: μ = 21
H₁: μ ≠ 21

A random sample of size 22 has a sample mean x = 19 from a population with standard deviation σ = 6.

(a) What is the value of c = 1 − α?
0.99

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	15.70
upper limit	22.30

What is the value of μ given in the null hypothesis (i.e., what is k)?
k =21

Is this value in the confidence interval?

Yes  

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.)
P=0.1179

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.    

Confidence Interval Estimate for the Mean
Data
Population Standard Deviation	6
Sample Mean	19
Sample Size	22
Confidence Level	99%
Intermediate Calculations
Standard Error of the Mean	1.2792
Z Value	-2.5758
Interval Half Width	3.2950
Confidence Interval
Interval Lower Limit	15.7050
Interval Upper Limit	22.2950

Z Test of Hypothesis for the Mean
Data
Null Hypothesis                       m=	21
Level of Significance	0.01
Population Standard Deviation	6
Sample Size	22
Sample Mean	19
Intermediate Calculations
Standard Error of the Mean	1.2792
Z Test Statistic	-1.5635
Two-Tail Test
Lower Critical Value	-2.5758
Upper Critical Value	2.5758
p-Value	0.1179
Do not reject the null hypothesis