Question

Fifty percent of registered voters in a congressional district are registered Democrats. The Republican candidate takes a random poll to assess his chances in a two-candidate race. He polls 1200 potential voters and finds that 621 plan to vote for the Democratic candidate. a) Find the point estimate for the proportion of voters who plan to vote Democratic. b) Is the sample large enough? Show your work. c) Does the Republican candidate have a chance to win? Test the claim. Use α = 0.05 . Show the 6 steps of the test of hypothesis.

Answer 1

The point estimate for Democratic voters is 0.5175, the sample size is sufficient, and after a hypothesis test with α = 0.05, we fail to reject the null hypothesis, indicating the Republican candidate could have a chance of winning.

Step-by-step explanation:

To address the student's question regarding the congressional district voter preferences, we shall perform a hypothesis test and calculate the point estimate for the proportion of voters who plan to vote Democratic.

Part A: Point Estimate

The point estimate for the proportion of voters who plan to vote Democratic can be found by dividing the number of potential voters who indicated they would vote for the Democratic candidate by the total number of potential voters polled. Thus:

Point estimate = 621 / 1200 = 0.5175

Part B: Sample Size Sufficiency

The sample size is large enough if np(1 - p) ≥ 5, where n is the sample size and p is the estimated proportion. With 1200 voters polled and an estimated proportion of 0.5175:

1200 * 0.5175 * (1 - 0.5175) = 1200 * 0.5175 * 0.4825 ≈ 300, which is much greater than 5, indicating the sample is sufficiently large.

Part C: Hypothesis Test

To determine if the Republican candidate has a chance to win, we use a hypothesis test:

1. Null Hypothesis (H0): p = 0.5 (The proportion of voters for the Democratic candidate is 50%)
2. Alternative Hypothesis (H1): p ≠ 0.5 (The proportion of voters for the Democratic candidate is not 50%)
3. Significance Level (α) = 0.05
4. Calculate the test statistic using the formula z = (p' - p)/√[p(1 - p)/n], where p' is the sample proportion and n is the sample size. This yields:
5. The critical value for α = 0.05 (two-tailed test) is approximately ±1.96
6. Make a decision: if the test statistic falls within the critical region, we reject H0. Otherwise, we fail to reject H0.

Based on the calculation:

z = (0.5175 - 0.5)/√[0.5(1 - 0.5)/1200] = 0.0175 / 0.0144 ≈ 1.215

Since the calculated z-value does not exceed the critical value of ±1.96, we fail to reject H0. This means with a significance level of 0.05, there is not enough evidence to say that the proportion is different from 50%. Therefore, it's possible that the Republican candidate has a chance of winning the election.

Answer 2

The point estimate for the proportion of voters who plan to vote Democratic is 0.5175, the sample is large enough
`(np ≥ 10 and nq ≥ 10)`, and the Republican candidate has a chance to win (fail to reject the null hypothesis).

How is that so?

a) Point estimate for the proportion of voters who plan to vote Democratic: