Question 1185823
Here's how to address each part of the problem:

**(a) Type I vs. Type II Error:**

* **Type I Error (False Positive):** Rejecting the null hypothesis (H0) when it is actually *true*.  In this context, it would mean concluding that the average height is different from a certain value when it actually *is* that value.

* **Type II Error (False Negative):** Failing to reject the null hypothesis (H0) when it is actually *false*.  In this context, it would mean failing to conclude that the average height is different from a certain value when it actually *is* different.

**(b) 95% Confidence Interval for μ:**

Since the population standard deviation (σ) is known, we use a z-interval:

```
Confidence Interval = x̄ ± z * (σ / √n)
```

For a 95% confidence level, the z-score is 1.96.

```
Confidence Interval = 70 ± 1.96 * (3 / √25)
Confidence Interval = 70 ± 1.96 * 0.6
Confidence Interval = 70 ± 1.176
Confidence Interval = (68.824, 71.176)
```

We are 95% confident that the true population mean height (μ) lies between 68.824 inches and 71.176 inches.

**(c) Hypothesis Test: H0: μ = 71 vs. H1: μ ≠ 71 (α = 0.05):**

1. **Test Statistic:**

   ```
   z = (x̄ - μ) / (σ / √n)
   z = (70 - 71) / (3 / √25)
   z = -1 / 0.6
   z ≈ -1.67
   ```

2. **Critical Value:** For a two-tailed test at α = 0.05, the critical z-values are ±1.96.

3. **Decision:** Since the calculated z-score (-1.67) falls *within* the range of -1.96 to +1.96, we *fail to reject* the null hypothesis.

4. **Conclusion:** There is not enough evidence at the 0.05 significance level to conclude that the population mean height is different from 71 inches.

**(d) Hypothesis Test: H0: μ = 72 vs. H1: μ ≠ 72 (α = 0.05):**

1. **Test Statistic:**

   ```
   z = (70 - 72) / (3 / √25)
   z = -2 / 0.6
   z ≈ -3.33
   ```

2. **Critical Value:** Same as above: ±1.96.

3. **Decision:** Since the calculated z-score (-3.33) falls *outside* the range of -1.96 to +1.96, we *reject* the null hypothesis.

4. **Conclusion:** There *is* sufficient evidence at the 0.05 significance level to conclude that the population mean height is different from 72 inches.

**(e) Hypothesis Test: H0: μ = 69 vs. H1: μ > 69 (α = 0.05):**

1. **Test Statistic:**

   ```
   z = (70 - 69) / (3 / √25)
   z = 1 / 0.6
   z ≈ 1.67
   ```

2. **Critical Value:** For a *one-tailed* (right-tailed) test at α = 0.05, the critical z-value is 1.645.

3. **Decision:** Since the calculated z-score (1.67) is *greater* than the critical value (1.645), we *reject* the null hypothesis.

4. **Conclusion:** There *is* sufficient evidence at the 0.05 significance level to conclude that the population mean height is greater than 69 inches.