Consider the following hypotheses.
H Subscript 0​: muμless than or equals≤208208
H Subscript A​: muμgreater than>208208
A sample is to be taken from a population with a mean of 209209 and a standard deviation of 22. The hypothesis test is to be conducted using a significance level of 0.050.05. Determine the probability of committing a Type II error for the conditions given in parts a through c.
a. nequals=1010
b. nequals=3030
c. nequals=5050
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Part 1
a. The probability of committing a Type II error is

In this problem, you're dealing with hypothesis testing, and you're tasked with calculating the probability of committing a **Type II error (β)** for different sample sizes. A **Type II error** occurs when you fail to reject the null hypothesis (\( H_0 \)) even though the alternative hypothesis (\( H_A \)) is true.

### Hypotheses:
- Null Hypothesis: \( H_0: \mu \leq 208 \)
- Alternative Hypothesis: \( H_A: \mu > 208 \)

Given:
- Population mean: \( \mu_0 = 209 \)
- Null hypothesis population mean: \( \mu = 208 \)
- Population standard deviation: \( \sigma = 22 \)
- Significance level: \( \alpha = 0.05 \)

Now, to calculate the probability of a Type II error, we follow these steps:

### Step-by-Step Process

1. **Calculate the critical value for \( \alpha = 0.05 \):**

   Since the hypothesis is one-tailed, we find the critical value of \( z \) corresponding to \( \alpha = 0.05 \) for a one-tailed test. Using a standard normal distribution table:
   
   \[
   z_{\alpha} = 1.645
   \]
   This means that if the test statistic exceeds this value, we reject \( H_0 \).

2. **Compute the standard error of the mean:**

   The standard error depends on the sample size \( n \):
   
   \[
   \text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{22}{\sqrt{n}}
   \]

3. **Find the critical value in terms of the sample mean:**

   Using the critical value \( z_{\alpha} = 1.645 \), we convert it into the corresponding value for the sample mean:
   
   \[
   \text{Critical Value for } \bar{x} = 208 + z_{\alpha} \cdot SE
   \]
   \[
   \text{Critical Value for } \bar{x} = 208 + 1.645 \cdot \frac{22}{\sqrt{n}}
   \]

4. **Find the probability of committing a Type II error:**

   Now, under the assumption that the true population mean \( \mu_0 = 209 \), we calculate the test statistic based on this mean:

   \[
   z_{\beta} = \frac{\text{Critical Value for } \bar{x} - 209}{SE}
   \]
   Then, look up the corresponding probability (or area) in the standard normal distribution table to get \( \beta \), the probability of Type II error.

### a. For \( n = 10 \):

- Standard Error: \( SE = \frac{22}{\sqrt{10}} \approx 6.96 \)
- Critical Value for \( \bar{x} \): \( 208 + 1.645 \cdot 6.96 \approx 219.45 \)

Now calculate the \( z \)-score for the true mean \( \mu = 209 \):

\[
z_{\beta} = \frac{219.45 - 209}{6.96} \approx 1.50
\]

Looking up \( z = 1.50 \) in the standard normal distribution table gives us a probability of approximately 0.9332. The area to the right of this is \( 1 - 0.9332 = 0.0668 \).

Thus, the probability of a Type II error for \( n = 10 \) is approximately \( \beta = 0.0668 \).

---

Would you like to proceed with parts b and c? Let me know if you want further clarification or a detailed explanation of any step!

Here are 5 follow-up questions to consider:
1. How does increasing the sample size affect the standard error and Type II error?
2. Why do we use the z-distribution in this hypothesis test?
3. How would the calculation differ if the population standard deviation were unknown?
4. What would happen if we decreased the significance level \( \alpha \) from 0.05 to 0.01?
5. Can you explain the relationship between Type I and Type II errors?

**Tip:** Increasing the sample size reduces the standard error, which makes it easier to detect smaller differences from the hypothesized mean, thus reducing the probability of a Type II error.

Consider the following hypotheses. H0: μ ≤ 208, HA: μ > 208. A sample is to be taken from a population with a mean of 209 and a standard deviation of 22. The hypothesis test is to be conducted using a significance level of 0.05. Determine the probability of committing a Type II error for the conditions given in parts a through c. a. n = 10, b. n = 30, c. n = 50.

Explore how to calculate the probability of committing a Type II error for different sample sizes (n = 10, 30, 50) in a hypothesis test where μ > 208, with a population mean of 209 and standard deviation of 22. The problem covers concepts of standard error, z-values, and the significance level of 0.05. Ideal for undergraduate statistics students.

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