A researcher claims that the proportion of people over 65 years of age in a certain city is 11%. To test this claim, a sample of 1000 people are taken and its determine that 126 people are over 65 years of age. The following is the setup for this hypothesis test: H0:p=0.11 Ha:p≠0.11 In this example, the p-value was determined to be 0.106. Come to a conclusion and interpret the results for this hypothesis test for a proportion (use a significance level of 5%) Select the correct answer below: The decision is to reject the Null Hypothesis. The conclusion is that there is enough evidence to reject the claim. The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to reject the claim.

We are given that a researcher claims that the proportion of people over 65 years of age in a certain city is 11%. To test this claim, a sample of 1000 people are taken and its determine that 126 people are over 65 years of age.

Let p = proportion of people over 65 years of age in a certain city = 11%

So, Null Hypothesis, [tex]H_0[/tex] : p = 0.11 {means that proportion of people over 65 years of age in a certain city is 11%}

Alternate Hypothesis, [tex]H_1[/tex] : p [tex]\neq[/tex] 0.11 {means that proportion of people over 65 years of age in a certain city is different from 11%}

The test statistics that will be used here is One-sample z proportion test;

T.S. = [tex]\frac{\hat p -p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }[/tex] ~ N(0,1)

where, [tex]\hat p[/tex] = proportion of people over 65 years of age in a sample of 1000 people = [tex]\frac{126}{1000}[/tex] = 0.126

n = sample of people = 1000

So, test statistics = [tex]\frac{0.126-0.11}{\sqrt{\frac{0.126(1-0.126)}{1000} } }[/tex]

= 1.525

Now, we are given with the P-value and the significance level in the question;

P-value = 0.106 or 10.6%

Significance level = 5%

Our decision rule based on p-value is as follows;

If P-value is less than the significance level, then we will reject our null hypothesis.

If P-value is more than the significance level, then we will not reject our null hypothesis.

Here, as we can clearly see that P-value is more than the significance level as 10.6% > 5%, so we have insufficient evidence to reject our null hypothesis.

Therefore, The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to reject the claim.