https://file.aiquickdraw.com/mathbot/file/17294556627162qfpiq11.jpg

The problem in the image is a hypothesis testing question using the P-value approach. You are asked to verify if the requirements of the test are satisfied by checking whether the product \( np_0(1 - p_0) \geq 10 \).

Let's break this down:

### Given:
- Null Hypothesis (\( H_0 \)): \( p = 0.74 \)
- Alternative Hypothesis (\( H_1 \)): \( p \neq 0.74 \)
- Sample size \( n = 500 \)
- Number of successes \( x = 360 \)
- Significance level \( \alpha = 0.01 \)
- \( p_0 = 0.74 \) (the hypothesized population proportion)

### Step 1: Check \( np_0(1 - p_0) \)
We need to calculate:

\[
np_0(1 - p_0) = 500 \times 0.74 \times (1 - 0.74) = 500 \times 0.74 \times 0.26
\]

### Step 2: Perform the Calculation

\[
np_0(1 - p_0) = 500 \times 0.74 \times 0.26 = 96.2
\]

### Step 3: Compare with 10
Since \( 96.2 \geq 10 \), the condition is satisfied.

Therefore, the correct answer would be **Yes**, because \( np_0(1 - p_0) = 96.2 \), which is greater than 10.

Let me know if you'd like further details on this!

Here are five related questions to expand your understanding:
1. How do you interpret the P-value in hypothesis testing?
2. What does it mean when a test fails to meet the requirement \( np_0(1 - p_0) \geq 10 \)?
3. How is the test statistic calculated for hypothesis testing for proportions?
4. What is the role of the significance level \( \alpha \) in hypothesis testing?
5. How do you compute the P-value for a two-tailed hypothesis test?

**Tip:** Always verify the assumptions and conditions of a hypothesis test before performing the calculations. This ensures the validity of the test results.

Test the hypothesis using the P-value approach. H0: p = 0.74 versus H1: p ≠ 0.74, n = 500, x = 360, α = 0.01. Is np0(1 - p0) ≥ 10? Select the correct choice below and fill in the answer box to complete your choice.

Learn how to test a hypothesis using the P-value approach for proportions. This example tests H0: p = 0.74 versus H1: p ≠ 0.74 with a sample size of 500, number of successes 360, and significance level α = 0.01. The np0(1 - p0) condition is verified as part of the requirements.

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