https://file.aiquickdraw.com/mathbot/file/1728505046156y80rlk9b.jpg

The image contains a hypothesis test scenario where the P-value approach is being used. Here's a breakdown of the information:

- **Null hypothesis \( H_0 \):** \( p = 0.59 \)
- **Alternative hypothesis \( H_1 \):** \( p < 0.59 \)
- **Sample size \( n \):** 150
- **Number of successes \( x \):** 81
- **Significance level \( \alpha \):** 0.01

Additionally, the condition to verify if the normal approximation can be used is checked:
- The condition \( np_0(1 - p_0) \geq 10 \) has been met.

The technology-based calculation of the **P-value** is provided:
- **P-value = 0.208**, rounded to three decimal places.

### Conclusion:
Given that the **P-value (0.208)** is greater than the **significance level \( \alpha = 0.01 \)**, we do **not reject the null hypothesis**. This means there is not enough evidence to conclude that the true proportion is less than 0.59 at the 1% significance level.

Would you like more details or clarifications? 

Here are 5 related questions to expand your understanding:
1. What is the interpretation of the P-value in hypothesis testing?
2. How is the test statistic calculated for a proportion hypothesis test?
3. Why is the condition \( np_0(1 - p_0) \geq 10 \) important for using the normal approximation?
4. How would the conclusion change if the significance level \( \alpha \) were higher (e.g., 0.05)?
5. What are the steps for manually calculating the P-value without technology?

**Tip:** Always compare the P-value with the significance level \( \alpha \) to decide whether to reject or fail to reject the null hypothesis.

Test the hypothesis using the P-value approach: H0: p = 0.59 versus H1: p < 0.59, n = 150, x = 81, α = 0.01.

Perform a hypothesis test for a population proportion using the P-value approach. The problem compares p = 0.59 against an alternative of p < 0.59 with n = 150 and x = 81, and calculates a P-value of 0.208 at a 0.01 significance level.

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