Question

In the last mayoral election in a large city, 47% of the adults over the age of 65 voted Republican. A researcher wishes to determine if the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election has changed. Let p represent the proportion of the population of all adults over the age of 65 in the city that plan to vote Republican in the next mayoral election. In terms of p, the researcher should test which of the following null and alternative hypotheses?H0: p = 0.47 vs. Ha: p > 0.47H0: p = 0.47 vs. Ha: p < 0.47H0: p = 0.47 vs. Ha: p ≠ 0.47

Answer 1

Null Hypothesis,
`H_0` : p = 47%

Alternate Hypothesis,
`H_A` : p
`\\eq` 47%

Explanation:

We are given that in the last mayoral election in a large city, 47% of the adults over the age of 65 voted Republican.

A researcher wishes to determine if the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election has changed.

Let p = proportion of the population of all adults over the age of 65 in the city that plan to vote Republican in the next mayoral election.

So, Null Hypothesis,
`H_0` : p = 47%

Alternate Hypothesis,
`H_A` : p
`\\eq` 47%

Here, null hypothesis states that the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election is equal to 47%.

On the other hand, alternate hypothesis states that the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election is different from 47%.

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

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

Hence, the researcher should test the above null and alternative hypotheses.