In a random sample of 400 ​items, 101 are defective. If the null hypothesis is that 24 % of the items in the population are​ defective, what is the value of Upper Z Subscript STAT​?

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.

Learn how to calculate the Z-Statistic for a hypothesis test on population proportion. This problem involves a sample of 400 items, with 101 defective, and tests against a null hypothesis of 24% defective. Step-by-step solution includes key concepts like sample proportion, standard error, and the Z-Statistic formula.

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