Question 1185823: Suppose in a sample of 25 people, the mean height XBAR was observed to be 70 inches. Suppose also SIGMA =3
(a) State the difference between type l error and type ll error
(b) Construct a 95% confidence interval for MU
(c) Would you reject the hypothesis H (0): MU = 71 versus (1): MU=/=71 on the basis of the observations, when testing at level ALPHA =0.05?
(d) Would you reject the hypothesis H (0): MU =72 versus the alternative H(1): MU=/=72 on the basis of the observations, when testing at level ALPHA=0.05?
(e) Would you reject the hypothesis H(0): MU =69 versus the (one-sided) alternative?
H(1): MU>69 on the basis of your observations, when testing at level ALPHA=0.05?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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