To find the value of $Z_{\text{STAT}}$, we can use the formula for the test statistic for a population proportion:

$Z_{\text{STAT}} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Where:

• $\hat{p}$ is the sample proportion of defective items.
• $p_0$ is the hypothesized population proportion (the null hypothesis value).
• $n$ is the sample size.

Step-by-Step Calculation:

1. Sample proportion $\hat{p}$: $\hat{p} = \frac{101}{400} = 0.2525$

2. Hypothesized proportion $p_0$: $p_0 = 0.24$

3. Sample size $n$: $n = 400$

4. Standard error of the proportion: $\text{SE} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.24(1 - 0.24)}{400}} = \sqrt{\frac{0.24 \times 0.76}{400}} = \sqrt{0.000456} \approx 0.02137$

5. Calculate $Z_{\text{STAT}}$: $Z_{\text{STAT}} = \frac{0.2525 - 0.24}{0.02137} \approx \frac{0.0125}{0.02137} \approx 0.585$

Thus, the value of $Z_{\text{STAT}}$ is approximately 0.585.

Would you like more details, or do you have any questions?

Follow-up Questions:

1. What is the significance level for a typical hypothesis test involving proportions?
2. How do we interpret the $Z_{\text{STAT}}$ value in relation to the null hypothesis?
3. What is the rejection region for a two-tailed hypothesis test at the 5% significance level?
4. How can you calculate the p-value corresponding to the $Z_{\text{STAT}}$ value?
5. What are Type I and Type II errors in the context of hypothesis testing?

Tip: Always compare the $Z_{\text{STAT}}$ value to critical values from the standard normal distribution when conducting hypothesis tests.