Step-by-step Hypothesis Testing Process:

We are tasked with determining if the proportion of spam emails in a company's sample is less than 77%. Here's how we can approach it:

1. State the Null and Alternative Hypotheses:

• Null hypothesis (H₀): The proportion of spam emails is greater than or equal to 77% (0.77).

  $H_0: p \geq 0.77$

• Alternative hypothesis (H₁): The proportion of spam emails is less than 77% (0.77).

  $H_1: p < 0.77$

This is a left-tailed test because we are testing whether the proportion is less than 0.77.

2. Significance Levels (α):

We are asked to use two significance levels:

• $\alpha = 0.05$
• $\alpha = 0.10$

3. Sample Proportion (p̂):

We are given:

• Number of spam emails in the sample = 361
• Sample size $n = 500$

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

$\hat{p} = \frac{361}{500} = 0.722$

4. Test Statistic:

The test statistic for a proportion follows a z-distribution and is calculated as:

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

Where:

• $\hat{p} = 0.722$ (sample proportion)
• $p_0 = 0.77$ (hypothesized population proportion)
• $n = 500$ (sample size)

Substituting the values:

$z = \frac{0.722 - 0.77}{\sqrt{\frac{0.77(1 - 0.77)}{500}}} = \frac{-0.048}{\sqrt{\frac{0.77(0.23)}{500}}}$

$z = \frac{-0.048}{\sqrt{\frac{0.1771}{500}}} = \frac{-0.048}{\sqrt{0.0003542}} = \frac{-0.048}{0.01882} \approx -2.55$

5. Critical Value Method:

Now we compare the test statistic $z = -2.55$ to the critical values for each significance level.

• For $\alpha = 0.05$, the critical value for a left-tailed test is $z_{\alpha} = -1.645$.
• For $\alpha = 0.10$, the critical value is $z_{\alpha} = -1.28$.

6. Decision:

• At $\alpha = 0.05$, since $z = -2.55 < -1.645$, we reject the null hypothesis.
• At $\alpha = 0.10$, since $z = -2.55 < -1.28$, we reject the null hypothesis.

7. Conclusion:

There is sufficient evidence to conclude, at both the 5% and 10% significance levels, that the proportion of spam emails in this company is less than 77%.

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Would you like a detailed explanation of any specific part? Here are 5 follow-up questions you could explore:

1. How do we interpret the significance levels in hypothesis testing?
2. What is the role of the sample size in determining the test statistic?
3. How does the choice of one-tailed vs. two-tailed tests impact the critical value?
4. What are Type I and Type II errors in the context of this problem?
5. How would the conclusion change if we used a higher or lower significance level?

Tip: In hypothesis testing, always ensure that the test type (left, right, or two-tailed) matches the hypothesis you're testing to avoid incorrect conclusions.