How to get the test statistic for a sample

Question

Test the following claim (h1) at a significance level of alpha=0.05. H0:p1=p2. H1:p1<p2. you obtain 10.8% successes in a sample of size n1=701 from the first population. You obtain 12.3% successes in a sample of size n1=538 from the second population. What is the test statistic for this sample? What is the p-value for this sample?

Answer 1

To test the hypothesis \( H_0: p_1 = p_2 \) against \( H_1: p_1 < p_2 \) using the given sample data, we follow these steps:

1. **Calculate the sample proportions:**

\[

\hat{p}_1 = 0.108

\]

\[

\hat{p}_2 = 0.123

\]

2. **Calculate the pooled proportion:**

\[

\hat{p} = \frac{0.108 \times 701 + 0.123 \times 538}{701 + 538} = \frac{75.708 + 66.174}{1239} = \frac{141.882}{1239} \approx 0.1145

\]

3. **Calculate the standard error:**

\[

SE = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.1145 \times 0.8855 \left(\frac{1}{701} + \frac{1}{538}\right)}

\]

\[

SE = \sqrt{0.1145 \times 0.8855 \left(0.001426 + 0.001858\right)} = \sqrt{0.1014 \times 0.003284} \approx 0.0182

\]

4. **Calculate the test statistic (Z):**

\[

Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.108 - 0.123}{0.0182} \approx \frac{-0.015}{0.0182} \approx -0.824

\]

5. **Find the p-value:**

\[

\text{p-value} \approx 0.205

\]

Since the p-value (0.205) is greater than the significance level (\(\alpha = 0.05\)), we fail to reject the null hypothesis \( H_0: p_1 = p_2 \). This means that there is not enough evidence to support the claim that \( p_1 < p_2 \).

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