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In this problem, you're conducting a hypothesis test for a population proportion, specifically to determine whether the percentage of spam emails is greater than 70%. Let's break down how to construct the null and alternative hypotheses.

### Part (a): State the appropriate null and alternative hypotheses

1. **Null Hypothesis \( H_0 \)**: The null hypothesis always represents the status quo or assumption that there is no effect or change. In this case, the null hypothesis is that the true proportion of spam emails is equal to 70%. Mathematically, you would state:
   \[
   H_0: p = 0.70
   \]

2. **Alternative Hypothesis \( H_1 \)**: The alternative hypothesis is what you're testing for—in this case, that the true proportion of spam emails is greater than 70%. Mathematically:
   \[
   H_1: p > 0.70
   \]

### Type of Test
Since you're testing whether the proportion is **greater than** a certain value (one-sided test), this is a **right-tailed test**.

To summarize:

- \( H_0: p = 0.70 \)
- \( H_1: p > 0.70 \)
- This is a **right-tailed test**.

Would you like to proceed with calculating the test statistic and p-value, or do you have any further questions?

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Here are 5 questions to extend your understanding:

1. How do you calculate the test statistic for a hypothesis test about proportions?
2. What is the formula for the standard error of a proportion?
3. How does the significance level (α) affect the decision to reject or fail to reject the null hypothesis?
4. What does the p-value represent in a hypothesis test?
5. How would you interpret the results if \( p \) is not greater than 0.70 at both significance levels (α = 0.10 and α = 0.01)?

**Tip**: For proportion tests, use the Z-test when the sample size is large enough to approximate the binomial distribution with a normal distribution.

A researcher reported that 71.8% of all email sent in a recent month was spam. A system manager at a large corporation believes that the percentage at his company may be 70%. He examines a random sample of 500 emails received at an email server and finds that 374 of the messages are spam. Can you conclude that the percentage of emails that are spam is greater than 70%? Use both α = 0.10 and α = 0.01 levels of significance and the P-value method with the TI-84 Plus calculator.

Learn how to perform a hypothesis test for a population proportion to determine if the spam rate is greater than 70%. Using a sample of 500 emails with 374 identified as spam, this problem guides through formulating hypotheses, using a Z-test for proportions, and interpreting results at α = 0.10 and α = 0.01 significance levels. Suitable for high school statistics students in Grades 10-12.

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