To test the claim about the difference between two population proportions [tex]\( p_1 \)[/tex] and [tex]\( p_2 \)[/tex] using the sample statistics, we can use a hypothesis test for the difference between two proportions.
Given:
- [tex]\( x_1 = 22 \)[/tex], [tex]\( n_1 = 121 \)[/tex]
- [tex]\( x_2 = 91 \)[/tex], [tex]\( n_2 = 220 \)[/tex]
- Significance Level: [tex]\( \alpha = 0.05 \)[/tex]
- Claim: [tex]\( p_1 = p_2 \)[/tex]
Step-by-Step Solution:
1. State the Hypotheses:
Null Hypothesis ([tex]\( H_0 \)[/tex]):
[tex]\( p_1 = p_2 \)[/tex]
Alternative Hypothesis ([tex]\( H_a \)[/tex]):
[tex]\( p_1 \neq p_2 \)[/tex]
2. Calculate the sample proportions:
[tex]\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{22}{121} \approx 0.1818
\][/tex]
[tex]\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{91}{220} \approx 0.4136
\][/tex]
3. Calculate the pooled sample proportion:
The pooled proportion [tex]\( \hat{p}_{\text{pool}} \)[/tex] is a weighted average of the sample proportions:
[tex]\[
\hat{p}_{\text{pool}} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{22 + 91}{121 + 220} \approx 0.3314
\][/tex]
4. Calculate the standard error of the difference in sample proportions:
The standard error (SE) is given by:
[tex]\[
SE = \sqrt{\hat{p}_{\text{pool}} \left( 1 - \hat{p}_{\text{pool}} \right) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}
\][/tex]
Plugging in the values:
[tex]\[
SE = \sqrt{0.3314 \left( 1 - 0.3314 \right) \left( \frac{1}{121} + \frac{1}{220} \right)} \approx 0.0533
\][/tex]
5. Calculate the z-statistic:
The z-statistic measures how many standard errors the sample proportion difference is away from 0:
[tex]\[
z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.1818 - 0.4136}{0.0533} \approx -4.3513
\][/tex]
6. Determine the critical z-value for the given significance level:
For a two-tailed test with [tex]\( \alpha = 0.05 \)[/tex], the critical z-value is [tex]\( \pm 1.96 \)[/tex]. This comes from the standard normal distribution where 0.025 of the distribution lies in each tail.
7. Make the Decision:
Compare the absolute value of the calculated z-statistic with the critical z-value:
[tex]\[
|z| = 4.3513
\][/tex]
Since [tex]\( 4.3513 > 1.96 \)[/tex], we reject the null hypothesis.
Conclusion:
At the [tex]\( \alpha = 0.05 \)[/tex] significance level, there is sufficient evidence to reject the null hypothesis that the two population proportions are equal. Hence, we conclude that there is a significant difference between the population proportions [tex]\( p_1 \)[/tex] and [tex]\( p_2 \)[/tex].
