For two population proportions P1 and P2, you are testing the hypothesis H0: P1 = P2 versus H1: P1 > P2. Suppose you find sample proportions p1=0.40 and p2=0.35 with a sample of size N=100 of independent data from each population.

Verify that the assumptions for using the z-test hold here.
Calculate the z-statistic for the hypothesis test.
Calculate the critical value at a significance level of 0.01 and at a level of 0.10.
Determine whether to accept or reject H0 at a significance level of 0.01 and at a level of 0.10.

### Step 1: Check the Assumptions for a Z-Test of Two Population Proportions  
For a **z-test for two population proportions**, the following conditions must hold:  
1. **Random Samples:** The samples from both populations are independently and randomly selected.
2. **Normality:** The sample size should be large enough for the sampling distribution of the difference in proportions to be approximately normal. Specifically, we need:  
   - \(n_1 p_1 \geq 5\)  
   - \(n_1 (1 - p_1) \geq 5\)  
   - \(n_2 p_2 \geq 5\)  
   - \(n_2 (1 - p_2) \geq 5\)  
   Where \(n_1\) and \(n_2\) are the sample sizes (100 each in this case).

Let's verify these conditions.

\[
n_1 = n_2 = 100, \quad p_1 = 0.40, \quad p_2 = 0.35
\]
- \(n_1 p_1 = 100 \times 0.40 = 40 \geq 5\)
- \(n_1 (1 - p_1) = 100 \times (1 - 0.40) = 100 \times 0.60 = 60 \geq 5\)
- \(n_2 p_2 = 100 \times 0.35 = 35 \geq 5\)
- \(n_2 (1 - p_2) = 100 \times (1 - 0.35) = 100 \times 0.65 = 65 \geq 5\)

Since all the conditions hold, the assumptions are met.

---

### Step 2: Calculate the Z-Statistic  
We are testing:

\[
H_0: P_1 = P_2 \quad \text{vs} \quad H_1: P_1 > P_2
\]

The test statistic for comparing two proportions is given by:

\[
z = \frac{(p_1 - p_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}
\]

where:

\[
\hat{p} = \frac{n_1 p_1 + n_2 p_2}{n_1 + n_2} = \frac{100 \times 0.40 + 100 \times 0.35}{200} = \frac{40 + 35}{200} = 0.375
\]

Now, substitute into the formula:

\[
z = \frac{0.40 - 0.35}{\sqrt{0.375 \times (1 - 0.375) \left(\frac{1}{100} + \frac{1}{100}\right)}}
\]

\[
z = \frac{0.05}{\sqrt{0.375 \times 0.625 \times 0.02}}
\]

\[
z = \frac{0.05}{\sqrt{0.0046875}} = \frac{0.05}{0.06846} \approx 0.73
\]

So, the **z-statistic** is approximately:

\[
z = 0.73
\]

---

### Step 3: Calculate the Critical Values  
Since this is a **one-tailed test** (we are testing if \(P_1 > P_2\)), we need the **critical z-values** for two different significance levels:  
- **At \(\alpha = 0.01\):** Find the critical value \(z_{\alpha = 0.01}\) for a one-tailed test.
  - From z-tables, \(z_{0.01} \approx 2.33\).
  
- **At \(\alpha = 0.10\):** Find the critical value \(z_{\alpha = 0.10}\) for a one-tailed test.
  - From z-tables, \(z_{0.10} \approx 1.28\).

---

### Step 4: Decision to Accept or Reject \(H_0\)  
We compare the calculated z-statistic (\(z = 0.73\)) with the critical values at each significance level.

- **At \(\alpha = 0.01\):**  
  - Critical value: \(z_{0.01} = 2.33\)
  - Since \(0.73 < 2.33\), **we fail to reject \(H_0\)** at the 1% significance level.

- **At \(\alpha = 0.10\):**  
  - Critical value: \(z_{0.10} = 1.28\)
  - Since \(0.73 < 1.28\), **we fail to reject \(H_0\)** at the 10% significance level.

---

### Conclusion  
At both significance levels (\(\alpha = 0.01\) and \(\alpha = 0.10\)), we **fail to reject the null hypothesis \(H_0: P_1 = P_2\)**. This suggests that there is not enough evidence to conclude that \(P_1\) is greater than \(P_2\).

---

### Related Questions  
1. How do we interpret the p-value in hypothesis testing?
2. How is the pooled proportion used in two-proportion z-tests?
3. What are the differences between one-tailed and two-tailed hypothesis tests?
4. How do Type I and Type II errors relate to significance levels?
5. What alternatives to the z-test exist for small sample sizes?

---

**Tip:** A smaller p-value provides stronger evidence against the null hypothesis, making the choice of significance level critical in decision-making.

For two population proportions P1 and P2, you are testing the hypothesis H0: P1 = P2 versus H1: P1 > P2. Suppose you find sample proportions p1=0.40 and p2=0.35 with a sample of size N=100 of independent data from each population. Verify that the assumptions for using the z-test hold here. Calculate the z-statistic for the hypothesis test. Calculate the critical value at a significance level of 0.01 and at a level of 0.10. Determine whether to accept or reject H0 at a significance level of 0.01 and at a level of 0.10.

Perform a Z-test for two population proportions with sample sizes of 100 each, testing H0: P1 = P2 versus H1: P1 > P2. This problem involves calculating the z-statistic, verifying normality assumptions, and determining critical values at significance levels of 0.01 and 0.10. Learn the step-by-step process with detailed formulas and conditions.

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