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.