Question 1193365: A random sample of n=48 is to be used to test the hypothesis that the mean of a normal population with standard deviation σ = 7 is equal to 54 against the alternative hypothesis that it is less than 54. If that hypothesis is to be rejected if and only if the mean of the random sample is less than x̄ < 49, find:
(a) the probability of committing a Type I error.
(b) the probability of committing a Type II error when μ = 45.
(a) P(Type I Error) =
(b) P(Type II Error) =
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Define**
* **Type I Error:** Rejecting the null hypothesis (H0: μ = 54) when it is actually true.
* **Type II Error:** Failing to reject the null hypothesis (H0: μ = 54) when the alternative hypothesis (H1: μ < 54) is true.
**2. (a) Probability of Type I Error**
* **Find the z-score corresponding to the critical value (x̄ = 49):**
* z = (x̄ - μ) / (σ / √n)
* z = (49 - 54) / (7 / √48)
* z = -5 / (7 / 6.928)
* z ≈ -4.94
* **Find the probability of getting a z-score less than -4.94:**
* Using a standard normal distribution table or a calculator, we find P(Z < -4.94) is extremely small (approximately 0).
* **Therefore, the probability of committing a Type I error is approximately 0.**
**3. (b) Probability of Type II Error (when μ = 45)**
* **Find the z-score corresponding to the critical value (x̄ = 49) under the alternative hypothesis (μ = 45):**
* z = (x̄ - μ) / (σ / √n)
* z = (49 - 45) / (7 / √48)
* z = 4 / (7 / 6.928)
* z ≈ 3.94
* **Find the probability of getting a z-score greater than 3.94 under the alternative hypothesis:**
* P(Z > 3.94) = 1 - P(Z < 3.94)
* Using a standard normal distribution table or a calculator, we find P(Z < 3.94) is very close to 1.
* Therefore, P(Z > 3.94) is very small (approximately 0).
* **Therefore, the probability of committing a Type II error when μ = 45 is approximately 0.**
**In Summary:**
* **(a) P(Type I Error) ≈ 0**
* **(b) P(Type II Error) ≈ 0**
**Note:** These probabilities are very small due to the large sample size (n = 48) and the significant difference between the hypothesized mean (μ = 54) and the alternative mean (μ = 45).
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