Question

In order to estimate the proportion of all likely voters who will likely vote for the incumbent in the upcoming city’s mayoral race, a random sample of 267 likely voters is taken, finding that 65% state they will likely vote for the incumbent.​ The polling agency wishes to test whether there is evidence that more than 50% of likely voters will likely vote for the incumbent. Evaluate the strength of evidence for this hypothesis.​

Answer 1

Yes, there is evidence that more than 50% of likely voters will likely vote for the incumbent.

Explanation:

We are given that in order to estimate the proportion of all likely voters who will likely vote for the incumbent in the upcoming city’s mayoral race, a random sample of 267 likely voters is taken, finding that 65% state they will likely vote for the incumbent.​

The polling agency wishes to test whether there is evidence that more than 50% of likely voters will likely vote for the incumbent.

Let p = proportion of voters who will likely vote for the incumbent

SO, Null Hypothesis,
`H_0` : p
`\leq` 50% {means that less than or equal to 50% of likely voters will likely vote for the incumbent}

Alternate Hypothesis,
`H_A` : p > 50% {means that more than 50% of likely voters will likely vote for the incumbent}

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)

where,
`\hat p` = sample proportion of voters who will likely vote for the incumbent in a sample of 267 voters = 65% or 0.65

n = sample of voters = 267

So, test statistics =
`\frac{0.65-0.50}{{\sqrt{(0.65(1-0.65))/(267) } } } }`

= 5.139

Since in the question we are not given the level of significance so we assume it to be 5%. Now at 0.05 significance level, the z table gives critical value of 1.6449 for right-tailed test. Since our test statistics is more than the critical value of z so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the more than 50% of likely voters will likely vote for the incumbent. The strength of the evidence is 95%.