Spam: 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 a=.10  and a=.01 levels of significance and the P-value method with the TI-84 Plus calculator.

To determine whether the percentage of spam emails is greater than 70% using the sample data and the given significance levels, we can perform a one-proportion z-test. Here are the steps:

### Step 1: State the Hypotheses
- Null Hypothesis (\(H_0\)): \(p = 0.70\)
- Alternative Hypothesis (\(H_a\)): \(p > 0.70\)

### Step 2: Calculate the Test Statistic
Using the sample proportion:
- Sample size (\(n\)) = 500
- Number of spam emails (\(x\)) = 374
- Sample proportion (\(\hat{p}\)) = \(\frac{374}{500} = 0.748\)

The test statistic for a one-proportion z-test is calculated using:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Where:
- \(\hat{p}\) is the sample proportion
- \(p_0\) is the population proportion under the null hypothesis (0.70)
- \(n\) is the sample size

Plugging in the values:
\[ z = \frac{0.748 - 0.70}{\sqrt{\frac{0.70(1 - 0.70)}{500}}} = \frac{0.048}{\sqrt{\frac{0.21}{500}}} = \frac{0.048}{\sqrt{0.00042}} = \frac{0.048}{0.02049} \approx 2.34 \]

### Step 3: Calculate the P-value
Using the z-table or the TI-84 Plus calculator, we can find the P-value corresponding to the calculated z-value of 2.34.

On the TI-84 Plus:
1. Press `2nd` and then `VARS` to access the distribution menu.
2. Choose `normalcdf`.
3. Enter `2.34, 1E99` (since we are interested in the area to the right of 2.34).

The P-value is approximately 0.0096.

### Step 4: Compare the P-value with the Significance Levels
- At \(\alpha = 0.10\):
  - Since the P-value (0.0096) is less than 0.10, we reject the null hypothesis.
- At \(\alpha = 0.01\):
  - Since the P-value (0.0096) is less than 0.01, we reject the null hypothesis.

### Conclusion
At both \(\alpha = 0.10\) and \(\alpha = 0.01\) levels of significance, we reject the null hypothesis. This means that there is sufficient evidence to conclude that the percentage of spam emails is greater than 70%.

Would you like further details or have any questions?

Here are five questions you might find interesting:
1. How does the sample size affect the outcome of a hypothesis test?
2. What is the difference between the P-value method and the critical value method in hypothesis testing?
3. How do you interpret the results of a hypothesis test in the context of the problem?
4. What are the assumptions made when performing a one-proportion z-test?
5. How can you use the TI-84 Plus calculator for different types of hypothesis tests?

**Tip:** When conducting hypothesis tests, always ensure that the sample size is sufficiently large to satisfy the assumptions of the test, particularly the normality assumption for the z-test.

Explore hypothesis testing to determine if the percentage of spam emails is greater than 70%. Utilize a one-proportion z-test with sample data and significance levels (alpha) of 0.10 and 0.01. Detailed steps and calculations provided.

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