https://file.aiquickdraw.com/mathbot/file/1726550189941wdp8hshk.jpg

Let's work through the hypothesis tests presented in the image.

### 10. Hypothesis Test
**Hypotheses:**
- \( H_0: \mu \leq 25 \)
- \( H_a: \mu > 25 \)

Given:
- Sample size (\(n\)) = 40
- Sample mean (\(\bar{x}\)) = 26.4
- Population standard deviation (\(\sigma\)) = 6

We will perform a one-sample z-test.

**a. Compute the value of the test statistic (z):**

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 26.4\) (sample mean),
- \(\mu_0 = 25\) (hypothesized population mean),
- \(\sigma = 6\) (population standard deviation),
- \(n = 40\) (sample size).

Let's calculate the z-value.

\[
z = \frac{26.4 - 25}{\frac{6}{\sqrt{40}}}
\]

**b. What is the p-value?**

Once the z-value is calculated, the p-value is obtained by checking the cumulative distribution for that z-value in a standard normal distribution table.

**c. At \(\alpha = 0.01\), what is your conclusion?**

We'll compare the p-value with \(\alpha = 0.01\). If \(p < 0.01\), reject \(H_0\); otherwise, do not reject \(H_0\).

**d. What is the rejection rule using the critical value?**

The critical value for \(\alpha = 0.01\) for a one-tailed z-test can be found using the z-distribution table. Compare the calculated z-value to this critical value.

---

### 11. Hypothesis Test
**Hypotheses:**
- \( H_0: \mu = 15 \)
- \( H_a: \mu \neq 15 \)

Given:
- Sample size (\(n\)) = 50
- Sample mean (\(\bar{x}\)) = 14.15
- Population standard deviation (\(\sigma\)) = 3

This is a two-tailed z-test.

**a. Compute the value of the test statistic (z):**

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 14.15\) (sample mean),
- \(\mu_0 = 15\),
- \(\sigma = 3\),
- \(n = 50\).

**b. What is the p-value?**

Find the p-value based on the z-value.

**c. At \(\alpha = 0.05\), what is your conclusion?**

Compare the p-value to \(\alpha = 0.05\) for decision-making.

**d. What is the rejection rule using the critical value?**

For a two-tailed test with \(\alpha = 0.05\), find the critical z-value (±1.96) and compare the calculated z-value.

---

### 12. Hypothesis Test
**Hypotheses:**
- \( H_0: \mu \geq 80 \)
- \( H_a: \mu < 80 \)

Given:
- Sample size (\(n\)) = 100
- Population standard deviation (\(\sigma\)) = 12

For the given sample means (\(\bar{x} = 78.5, 77, 75.5, 81\)), we will calculate the p-value and compare it to \(\alpha = 0.01\).

For each sample mean:
1. Calculate the z-value:
   \[
   z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
   \]
2. Find the p-value and state the conclusion for each.

---

Would you like me to compute the z-values and p-values for each of these cases?

Consider the hypothesis tests for the following problems: 
1. Hypotheses: H0: μ ≤ 25, Ha: μ > 25. A sample of 40 provided a sample mean of 26.4. The population standard deviation is 6. 
2. Hypotheses: H0: μ = 15, Ha: μ ≠ 15. A sample of 50 provided a sample mean of 14.15. The population standard deviation is 3. 
3. Hypotheses: H0: μ ≥ 80, Ha: μ < 80. A sample of 100 with population standard deviation 12 is used. Sample means are: 78.5, 77, 75.5, 81.

Learn how to perform hypothesis tests using the Z-test for various sample sizes and means. This guide covers key statistical concepts such as P-value, critical value, and decision-making based on α = 0.01 or 0.05 significance levels.

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