To determine whether the proportion of spam emails differs from 71%, we conduct a hypothesis test for a population proportion. Here's how the test proceeds:

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Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): $p = 0.71$ (The proportion of spam emails is 71%)
• Alternative Hypothesis ($H_a$): $p \neq 0.71$ (The proportion of spam emails is different from 71%)

This is a two-tailed test.

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Step 2: Test Statistic Formula

The test statistic for a proportion is:

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

Where:

• $\hat{p}$ = sample proportion = $\frac{x}{n} = \frac{380}{500} = 0.76$
• $p_0$ = hypothesized proportion = 0.71
• $n$ = sample size = 500

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Step 3: Calculate the Test Statistic

First, compute the standard error:

$\text{Standard Error} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.71(1-0.71)}{500}} = \sqrt{\frac{0.71 \cdot 0.29}{500}} = \sqrt{0.0004118} \approx 0.0203$

Now, compute $z$:

$z = \frac{\hat{p} - p_0}{\text{Standard Error}} = \frac{0.76 - 0.71}{0.0203} = \frac{0.05}{0.0203} \approx 2.46$

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Step 4: Determine the Critical Values

For a two-tailed test:

• At $\alpha = 0.10$: Critical values are $\pm 1.645$
• At $\alpha = 0.01$: Critical values are $\pm 2.576$

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Step 5: Decision Rule

• If $z$-value falls outside the critical region, reject $H_0$.

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Step 6: Compare Test Statistic to Critical Values

1. At $\alpha = 0.10$:
   • $z = 2.46$ is greater than $1.645$, so we reject $H_0$.
2. At $\alpha = 0.01$:
   • $z = 2.46$ is less than $2.576$, so we fail to reject $H_0$.

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Conclusion

• At $\alpha = 0.10$: There is sufficient evidence to conclude that the proportion of spam emails differs from 71%.
• At $\alpha = 0.01$: There is insufficient evidence to conclude that the proportion of spam emails differs from 71%.

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Would you like further clarification or details on this test?
Here are 5 related questions to consider:

1. How would the conclusion change if the sample size were larger or smaller?
2. What if the observed proportion was 0.73 instead of 0.76? How would it affect the test?
3. Can this method be extended to test other population proportions or confidence intervals?
4. How does changing the significance level affect the likelihood of rejecting the null hypothesis?
5. What are the assumptions underlying the z-test for proportions?

Tip: Ensure your sample size is sufficiently large for the z-test to be valid; this is usually met if $n \cdot p_0 \geq 10$ and $n \cdot (1 - p_0) \geq 10$.