	a. Given the following hypothesis, determine the p-value when x ̅ = 21, n = 25, and σ = 5
H0: µ = 20
H1: µ ≠ 20
	b. Repeat part (a) with x ̅ = 22.
	c. Repeat part (a) with x ̅ = 23.
	d. Describe what happens to the value of the test statistic and its p-value when the value of x ̅ increases. 


Let's go through each part of the problem step by step.

### Part (a) Given:
- \( \bar{x} = 21 \) (Sample mean)
- \( n = 25 \) (Sample size)
- \( \sigma = 5 \) (Population standard deviation)
- \( H_0: \mu = 20 \) (Null hypothesis)
- \( H_1: \mu \neq 20 \) (Alternative hypothesis)

**Step 1: Calculate the test statistic (z-value):**

The formula for the z-test statistic is:

\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]

Substituting the values:

\[
z = \frac{21 - 20}{\frac{5}{\sqrt{25}}} = \frac{1}{\frac{5}{5}} = 1
\]

**Step 2: Find the p-value:**

Since this is a two-tailed test, we need to find the probability that \( Z \) is greater than the absolute value of the test statistic and then double it.

Using a standard normal distribution table or calculator:

\[
P(Z > 1) = 0.1587
\]

So, the p-value for a two-tailed test is:

\[
p = 2 \times 0.1587 = 0.3174
\]

### Part (b) Given:
- \( \bar{x} = 22 \)

Following the same steps as in part (a):

\[
z = \frac{22 - 20}{\frac{5}{\sqrt{25}}} = \frac{2}{1} = 2
\]

Using the standard normal distribution:

\[
P(Z > 2) = 0.0228
\]

So, the p-value is:

\[
p = 2 \times 0.0228 = 0.0456
\]

### Part (c) Given:
- \( \bar{x} = 23 \)

Again, following the same steps:

\[
z = \frac{23 - 20}{\frac{5}{\sqrt{25}}} = \frac{3}{1} = 3
\]

Using the standard normal distribution:

\[
P(Z > 3) = 0.0013
\]

So, the p-value is:

\[
p = 2 \times 0.0013 = 0.0026
\]

### Part (d) Analysis:

As the sample mean \( \bar{x} \) increases:
1. The test statistic (z-value) also increases. This is because the difference between \( \bar{x} \) and \( \mu \) increases, leading to a larger numerator in the z-value formula.
2. The p-value decreases. A larger z-value corresponds to a smaller probability in the tails of the normal distribution, hence a smaller p-value.

This suggests that as the sample mean moves further away from the hypothesized population mean, there is stronger evidence against the null hypothesis, reflected in a smaller p-value.

---

Do you want more details on any of these steps, or do you have any questions?

### Relative Questions:
1. What is the significance level, and how is it related to the p-value?
2. How does changing the sample size \( n \) affect the test statistic and p-value?
3. What would happen to the p-value if the population standard deviation \( \sigma \) were larger?
4. How do you interpret a p-value in the context of hypothesis testing?
5. How does the alternative hypothesis being one-tailed affect the p-value?
6. What is the difference between a z-test and a t-test, and when is each appropriate?
7. How would the p-value change if the sample mean \( \bar{x} \) were less than the population mean \( \mu \)?
8. What conclusions can be drawn if the p-value is greater than the significance level?

### Tip:
Always remember to check if your test should be one-tailed or two-tailed based on the alternative hypothesis.

Learn how to calculate p-values in hypothesis testing using different sample means with a detailed step-by-step solution. This problem covers key concepts in hypothesis testing, including the z-test statistic, normal distribution, and p-values. Suitable for college-level students studying statistics.

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