Question

The mayor of a town has proposed a plan for the annexation of a new community. A political study took a sample of 800 voters in the town and found that 50% of the residents favored annexation. Using the data, a political strategist wants to test the claim that the percentage of residents who favor annexation is less than 55% . Testing at the 0.02 level, is there enough evidence to support the strategist's claim?
Step 1 of 7: State the null and alternative hypotheses.
Step 2 of 7: Find the value of the test statistic. Round your answer to two decimal places.
Step 3 of 7: Specify if the test is one-tailed or two-tailed.
Step 4 of 7: Determine the P P -value of the test statistic. Round your answer to four decimal places.
Step 5 of 7: Identify the value of the level of significance.
Step 6 of 7: Make the decision to reject or fail to reject the null hypothesis.
Step 7 of 7: State the conclusion of the hypothesis test.

Answer 1

Answer:

Step-by-step explanation:

Hello!

A survey was conducted to know the population's opinion about the mayor's proposed plan of the annexation of a new community.

Of a sample of 800 voters, 50% favored the annexation.

The political strategist thinks that less than 55% of the voters favor the annexation.

For me, the first step of every statistical exercise is to determine the study variable and its distribution.

X: Number of voters, out of 800, in favor of annexation.

This variable has a binomial distribution with parameters n and p.

The parameter of the study is the population proportion of voters that favor the annexation.

The statistical hypotheses are:

H₀: p ≥ 0.55

H₁: p < 0.55

α: 0.02

To test the population proportion the best statistic to use is the approximation to standard normal:

$Z= \frac{p'-p}{\sqrt{\frac{p(1-p)}{n} } }$≈N(0;1)

$Z_{H_0}= \frac{0.5-0.55}{\sqrt{\frac{0.55*0.45}{800} } } = -2.84$

This hypothesis test is one-tailed to the left, this means, that you will reject the null hypothesis to small values of the statistic Z.

The critical value is $Z_{\alpha } = Z_{0.02} = -2.054$

The p-value is one-tailed in the same direction as the test and you can calculate it as:

P(Z≤-2.84)= 0.002

The p-value: 0.002 is less than the significance level α: 0.02 ⇒ The decision is to reject the null hypothesis.

Using a significance level of 2% the decision is to reject the null hypothesis. Then you can conclude that the claim of the political strategist is correct and the population proportion of voters that favor the annexation is less than 55%.

I hope it helps!