To solve this hypothesis testing problem, we will go through each of the steps systematically.

Step-by-Step Solution

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$

We want to test if the proportion of college students who say they're interested in their district's election results is more than 65%.

1. Null hypothesis $H_0$: $p = 0.65$
2. Alternative hypothesis $H_1$: $p > 0.65$

This is a right-tailed test, as we are testing if the proportion is greater than the hypothesized value.

(b) Determine the type of test statistic to use

Since we are dealing with proportions, we will use a z-test for proportions.

(c) Find the value of the test statistic

To calculate the test statistic $z$, we use the formula:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Where:

• $\hat{p}$ is the sample proportion,
• $p_0$ is the hypothesized population proportion (0.65),
• $n$ is the sample size.

Given:

• Sample size $n = 255$
• Number of students interested $x = 176$

First, calculate the sample proportion $\hat{p}$:

$\hat{p} = \frac{x}{n} = \frac{176}{255} \approx 0.6902$

Now, plug these values into the formula for $z$:

$z = \frac{0.6902 - 0.65}{\sqrt{\frac{0.65 \times (1 - 0.65)}{255}}}$

Calculating the denominator:

$\sqrt{\frac{0.65 \times 0.35}{255}} \approx 0.0292$

So,

$z = \frac{0.6902 - 0.65}{0.0292} \approx \frac{0.0402}{0.0292} \approx 1.377$

Thus, the test statistic is approximately $z = 1.377$.

(d) Find the critical value

Since we are testing at the $\alpha = 0.01$ significance level for a right-tailed test, we need the critical value $z_{\alpha}$ for $\alpha = 0.01$.

Using the standard normal (z) distribution, the critical value at a 0.01 significance level for a one-tailed test is approximately:

$z_{\alpha} = 2.326$

(e) Conclusion

To determine if there is enough evidence to support the political scientist's claim, compare the test statistic to the critical value:

• If $z \geq z_{\alpha}$, reject $H_0$.
• If $z < z_{\alpha}$, do not reject $H_0$.

Here, $z = 1.377$ is less than the critical value of $2.326$. Therefore, we do not reject the null hypothesis.

Final Answer

There is not enough evidence to support the political scientist's claim that the proportion of college students interested in their district's election results is more than 65% at the 0.01 level of significance.

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Would you like more details on any specific part of this solution?

Related Questions

1. What changes if we test at a 0.05 level of significance instead of 0.01?
2. How would the solution change if it were a two-tailed test?
3. What are Type I and Type II errors in the context of this test?
4. How can we increase the power of this test?
5. What if the sample size were larger; how might that impact the conclusion?

Tip

For hypothesis tests involving proportions, always verify that $np$ and $n(1 - p)$ are both greater than 5 to ensure the normal approximation is appropriate.