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Let's tackle the problem systematically. Here's the breakdown for each question:

### Problem Summary
We are testing the hypothesis:
- Null Hypothesis (\(H_0\)): \(\pi_1 \leq \pi_2\)
- Alternative Hypothesis (\(H_1\)): \(\pi_1 > \pi_2\)

We are given:
- \(n_1 = 100\), \(X_1 = 70\), so \(\hat{p}_1 = \frac{70}{100} = 0.7\)
- \(n_2 = 150\), \(X_2 = 90\), so \(\hat{p}_2 = \frac{90}{150} = 0.6\)
- Significance level (\(\alpha\)) = 0.05.

Now let's solve step by step.

---

### a. State the Decision Rule
We use a **one-tailed z-test** since the hypothesis involves \(>\).

- Critical value (\(z_\alpha\)) at \(\alpha = 0.05\) for a one-tailed test is **1.645**.
- Decision Rule: Reject \(H_0\) if \(z_{\text{calculated}} > 1.645\).

---

### b. Compute the Pooled Proportion
The pooled proportion is calculated as:
\[
\hat{p} = \frac{X_1 + X_2}{n_1 + n_2} = \frac{70 + 90}{100 + 150} = \frac{160}{250} = 0.64
\]

---

### c. Compute the Test Statistic
The test statistic formula is:
\[
z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}
\]

Substitute the values:
\[
z = \frac{0.7 - 0.6}{\sqrt{0.64(1 - 0.64)\left(\frac{1}{100} + \frac{1}{150}\right)}}
\]

1. Compute the denominator:
   \[
   0.64(1 - 0.64) = 0.64 \times 0.36 = 0.2304
   \]
   \[
   \frac{1}{100} + \frac{1}{150} = 0.01 + 0.00667 = 0.01667
   \]
   \[
   0.2304 \times 0.01667 = 0.003841
   \]
   \[
   \sqrt{0.003841} = 0.062
   \]

2. Compute \(z\):
   \[
   z = \frac{0.7 - 0.6}{0.062} = \frac{0.1}{0.062} \approx 1.61
   \]

---

### d. Decision Regarding the Null Hypothesis
- \(z_{\text{calculated}} = 1.61\)
- Critical value \(z_\alpha = 1.645\)

Since \(z_{\text{calculated}} < 1.645\), **we fail to reject the null hypothesis**. There is not enough evidence to conclude that \(\pi_1 > \pi_2\).

---

Would you like further details or clarifications? Here are some follow-up questions for deeper exploration:

1. How does the choice of significance level impact the decision?
2. What changes if this were a two-tailed test instead of one-tailed?
3. How is the pooled proportion calculated and why is it important?
4. What are the implications of failing to reject the null hypothesis?
5. How can the sample sizes affect the reliability of the test?

**Tip:** Always verify critical values using a z-table or software to ensure accuracy.

Learn how to test hypotheses for proportions using a one-tailed z-test. This example explores testing H0: π1 ≤ π2 vs. H1: π1 > π2 with a sample size of n1=100 and n2=150. Includes detailed steps for decision rules, pooled proportion, test statistic, and conclusions.

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