Question

According to a​ research, 57.0​% of households nationwide used natural gas for heating during a year.​ Recently, a survey of 2,600 randomly selected households showed that 58.0​% used natural gas. Use a 0.05 significance level to test the claim that the 57.0​% national rate has changed.

Formulate the null and alternative hypotheses. Choose the correct answer below.

1.

A. H0​: p=0.57​, Ha​: p >0.57

B. H0​: p ≠0.57, Ha​: p=0.57

C. H0​: p > 0.57​, Ha​: p <0.57

D. H0​: p =0.57​, Ha​: p ≠0.57

2. Find the test statistic.

z =____

​(Round to one decimal place as​ needed.)

3. Find the​ P-value for the found test statistic.

​P-value=___

​(Round to four decimal places as​ needed.)

4.Now make a conclusion. Choose the correct answer below.

A.There is sufficient evidence to support the claim that the percentage of households using natural gas has changed.

B.There is not sufficient evidence to support the claim that the percentage of households using natural gas has changed.

C.There is not enough information to make a conclusion.

Answer 1

D. H0​: p =0.57​, Ha​: p ≠0.57

B.There is not sufficient evidence to support the claim that the percentage of households using natural gas has changed

Explanation:

given that according to a​ research, 57.0​% of households nationwide used natural gas for heating during a year.​ Recently, a survey of 2,600 randomly selected households showed that 58.0​% used natural gas.

Set up hypotheses as

D. H0​: p =0.57​, Ha​: p ≠0.57

(Two tailed test at 5% significance level)

Sample proportion = 0.58

sample size n = 2600

Std error of sample assuming H0 to be true =
`\sqrt{(0.57*0.43)/(2600) } \\=0.009362\\`

p difference =
`0.58-0.57 = 0.01`

Z statistic = p difference/std error = 1.068

p value = 0.2855

Since p > 0.05 we accept null hypothesis

Conclusion

p =0.57 has not changed.

B.There is not sufficient evidence to support the claim that the percentage of households using natural gas has changed