Questions: You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly different from 66% at a level of significance of a=0.01. According to your sample, 45 out of 73 potential voters prefer the Democratic candidate.

You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly different from 66% at a level of significance of a=0.01. According to your sample, 45 out of 73 potential voters prefer the Democratic candidate.

Transcript text: You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly different from 66% at a level of significance of $a=0.01$. According to your sample, 45 out of 73 potential voters prefer the Democratic candidate.

Solution

Solution Steps

To determine if the proportion of voters who prefer the Democratic candidate is significantly different from 66%, we can perform a hypothesis test for a population proportion. The steps are as follows:

State the Hypotheses:
- Null Hypothesis ($H_0$): $p = 0.66$
- Alternative Hypothesis ($H_a$): $p \neq 0.66$
Calculate the Test Statistic:
- Use the sample proportion ($\hat{p}$) and the population proportion under the null hypothesis ($p_0$) to calculate the z-score.
Determine the p-value:
- Use the z-score to find the p-value from the standard normal distribution.
Compare the p-value to the significance level ($\alpha = 0.01$):
- If the p-value is less than $\alpha$, reject the null hypothesis.

Step 1: State the Hypotheses

We are testing if the proportion of voters who prefer the Democratic candidate is significantly different from 66%.

Null Hypothesis ($H_0$): $p = 0.66$
Alternative Hypothesis ($H_a$): $p \neq 0.66$

Step 2: Calculate the Test Statistic

Given:

Sample size ($n$) = 73
Sample proportion ($\hat{p}$) = 0.6164
Population proportion ($p_0$) = 0.66
Significance level ($\alpha$) = 0.01

First, calculate the standard error (SE): \[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.66 \times (1 - 0.66)}{73}} = 0.05544 \]

Next, calculate the z-score: \[ z = \frac{\hat{p} - p_0}{SE} = \frac{0.6164 - 0.66}{0.05544} = -0.7857 \]

Step 3: Determine the p-value

Using the z-score, we find the p-value: \[ p\text{-value} = 2 \times (1 - \Phi(|z|)) = 2 \times (1 - \Phi(0.7857)) = 0.4320 \]

Step 4: Compare the p-value to the Significance Level

Compare the p-value to $\alpha$: \[ p\text{-value} = 0.4320 \quad \text{and} \quad \alpha = 0.01 \]

Since $p\text{-value} > \alpha$, we fail to reject the null hypothesis.

Final Answer

Based on the hypothesis test, we do not have sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is significantly different from 66%.

\[ \boxed{\text{The answer is d. There is a 43.2\% chance that the percent of all voters who prefer the Democratic candidate differs from 66\%.}} \]

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