Question

In October 1945, the Gallup organization asked 1,487 randomly sampled Americans, "Do you think we can develop a way to protect ourselves from atomic bombs in case other countries tried to use them against us?" Did a majority of Americans feel the United States could develop a way to protect itself from atomic bombs in 1945? Use the α = 0.05 level of significance.

Answer 1

Yes. There is enough evidence to support the claim that the proportion of Americans feel the United States could develop a way to protect itself from atomic bombs in 1945 is significantly greater than 0.5.

Explanation:

The question is incomplete:

In October 1945, the Gallup organization asked 1487 randomly sampled Americans, "Do you think we can develop a way to protect ourselves from atomic bombs in case other countries tried to use them against us?" with 788 responding yes. Did a majority of Americans feel the United States could develop a way to protect itself from atomic bombs in 1945? Use the α=0.05 level of significance.

This is a hypothesis test for a proportion.

The claim is that the proportion of Americans feel the United States could develop a way to protect itself from atomic bombs in 1945 is significantly greater than 0.5.

Then, the null and alternative hypothesis are:

H_0: \pi=0.5\\\\H_a:\pi>0.5

The significance level is 0.05.

The sample has a size n=1487.

The sample proportion is p=0.53.

`p=X/n=788/1487=0.53`

The standard error of the proportion is:

`\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.5*0.5)/(1487)}\\\\\\ \sigma_p=√(0.000168)=0.013`

Then, we can calculate the z-statistic as:

`z=(p-\pi-0.5/n)/(\sigma_p)=(0.53-0.5-0.5/1487)/(0.013)=(0.03)/(0.013)=2.288`

This test is a right-tailed test, so the P-value for this test is calculated as:

`\text{P-value}=P(z>2.288)=0.011`

As the P-value (0.011) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of Americans feel the United States could develop a way to protect itself from atomic bombs in 1945 is significantly greater than 0.5.