To determine if the percentage of spam emails differs from 77%, we can perform a hypothesis test for the population proportion. We will use the critical value method for both $\alpha = 0.01$ and $\alpha = 0.05$ levels of significance.

Step 1: State the Hypotheses

• Null hypothesis ($H_0$): $p = 0.77$ (The proportion of spam emails is 77%)
• Alternative hypothesis ($H_a$): $p \neq 0.77$ (The proportion of spam emails is not 77%)

Step 2: Calculate the Sample Proportion

The sample proportion ($\hat{p}$) is calculated as:

$\hat{p} = \frac{364}{500} = 0.728$

Step 3: Determine the Critical Values

Since this is a two-tailed test, we need to find the critical values for the standard normal distribution (Z-distribution):

• For $\alpha = 0.05$, the critical values are $Z_{\alpha/2} = \pm 1.96$
• For $\alpha = 0.01$, the critical values are $Z_{\alpha/2} = \pm 2.576$

Step 4: Calculate the Test Statistic

The test statistic for the population proportion is calculated using the formula:

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

where $p_0 = 0.77$ and $n = 500$.

$Z = \frac{0.728 - 0.77}{\sqrt{\frac{0.77 \times 0.23}{500}}}$

$Z = \frac{-0.042}{\sqrt{\frac{0.1771}{500}}}$

$Z = \frac{-0.042}{\sqrt{0.0003542}}$

$Z = \frac{-0.042}{0.01881} \approx -2.232$

Step 5: Make the Decision

• For $\alpha = 0.05$:

  • The critical region is $Z < -1.96$ or $Z > 1.96$.
  • Since $-2.232 < -1.96$, we reject the null hypothesis at $\alpha = 0.05$.

• For $\alpha = 0.01$:

  • The critical region is $Z < -2.576$ or $Z > 2.576$.
  • Since $-2.232$ does not fall in the critical region, we fail to reject the null hypothesis at $\alpha = 0.01$.

Conclusion

• At the $\alpha = 0.05$ significance level, there is enough evidence to conclude that the percentage of spam emails differs from 77%.
• At the $\alpha = 0.01$ significance level, there is not enough evidence to conclude that the percentage of spam emails differs from 77%.