Question

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

a) A poll of 1068 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At the 0.05 level of significance, test the claim that at least half of all voters prefer the Democrat.

b) In a sample of 167 children selected randomly from one town, it is found that 37 of them suffer from asthma. At the 0.05 significance level, test the claim that the proportion of all children in the town who suffer from asthma is 11%.

c) A nationwide study of American homeowners revealed that 64% have one or more lawn mowers. A lawn equipment manufacturer, located in Omaha, feels the estimate is too low for households in Omaha. Can the value 0.64 be rejected if a survey of 490 homes in Omaha yields 331 with one or more lawn mowers? Use  

Answer 1

SOLUTION A] (1) Null and Alternative Hypotheses: The following null and alternative hypotheses need to be tested:

Ho:p0.5

Ha:p0.5

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is zc​=-1.64.

The rejection region for this right-tailed test is R={z:z-1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis: Since it is observed that z=−1.285≤zc​= -1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:The P-Value is .100273.The result is not significant because p > .05.

(5) Conclusion: It is concluded that the null hypothesis Ho is not rejected.

b] (1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p=0.11

Ha:p≠​0.11

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=4.607>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0000, and since p=0.0000<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion: It is concluded that the null hypothesis Ho is rejected.

c] NULL HYPOTHESIS H0:

ALTERNATIVE HYPOTHESIS Ha:

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is zc​=1.64.

The rejection region for this left-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed tha z=1.638zc​=1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The P-Value is .050711.The result is not significant at p > .05.

(5) Conclusion: It is concluded that the null hypothesis Ho is not rejected.