Construct a 95% confidence interval for the difference between the two population proportions P1-P2. Population one has a sample size of 510 and a sample proportion of 25%. Population two has a sample size of 450 and a sample proportion of 19% (calculator).
The Correct Answer and Explanation is:
To construct a 95% confidence interval for the difference between two population proportions (P1 – P2), we can use the following formula:CI=(p1−p2)±Zα/2×p1(1−p1)n1+p2(1−p2)n2\text{CI} = (p_1 – p_2) \pm Z_{\alpha/2} \times \sqrt{\frac{p_1(1 – p_1)}{n_1} + \frac{p_2(1 – p_2)}{n_2}}CI=(p1−p2)±Zα/2×n1p1(1−p1)+n2p2(1−p2)
Where:
- p1p_1p1 and p2p_2p2 are the sample proportions for population one and population two, respectively.
- n1n_1n1 and n2n_2n2 are the sample sizes for population one and population two, respectively.
- Zα/2Z_{\alpha/2}Zα/2 is the critical value from the standard normal distribution for a 95% confidence level, which is approximately 1.96.
Step 1: Identify the given values
- Population one: n1=510n_1 = 510n1=510, p1=0.25p_1 = 0.25p1=0.25
- Population two: n2=450n_2 = 450n2=450, p2=0.19p_2 = 0.19p2=0.19
- Zα/2=1.96Z_{\alpha/2} = 1.96Zα/2=1.96 for a 95% confidence level.
Step 2: Calculate the standard error
We use the formula for the standard error of the difference between two proportions:SE=p1(1−p1)n1+p2(1−p2)n2SE = \sqrt{\frac{p_1(1 – p_1)}{n_1} + \frac{p_2(1 – p_2)}{n_2}}SE=n1p1(1−p1)+n2p2(1−p2)
Substituting the given values:SE=0.25(1−0.25)510+0.19(1−0.19)450=0.25×0.75510+0.19×0.81450SE = \sqrt{\frac{0.25(1 – 0.25)}{510} + \frac{0.19(1 – 0.19)}{450}} = \sqrt{\frac{0.25 \times 0.75}{510} + \frac{0.19 \times 0.81}{450}}SE=5100.25(1−0.25)+4500.19(1−0.19)=5100.25×0.75+4500.19×0.81SE=0.1875510+0.1539450=0.0003686+0.0003419=0.0007105≈0.0267SE = \sqrt{\frac{0.1875}{510} + \frac{0.1539}{450}} = \sqrt{0.0003686 + 0.0003419} = \sqrt{0.0007105} \approx 0.0267SE=5100.1875+4500.1539=0.0003686+0.0003419=0.0007105≈0.0267
Step 3: Calculate the confidence interval
Now, we can calculate the confidence interval:CI=(p1−p2)±Zα/2×SECI = (p_1 – p_2) \pm Z_{\alpha/2} \times SECI=(p1−p2)±Zα/2×SECI=(0.25−0.19)±1.96×0.0267CI = (0.25 – 0.19) \pm 1.96 \times 0.0267CI=(0.25−0.19)±1.96×0.0267CI=0.06±1.96×0.0267=0.06±0.0523CI = 0.06 \pm 1.96 \times 0.0267 = 0.06 \pm 0.0523CI=0.06±1.96×0.0267=0.06±0.0523
Thus, the confidence interval is:CI=(0.060−0.0523,0.060+0.0523)=(0.0077,0.1123)CI = (0.060 – 0.0523, 0.060 + 0.0523) = (0.0077, 0.1123)CI=(0.060−0.0523,0.060+0.0523)=(0.0077,0.1123)
Conclusion
The 95% confidence interval for the difference between the two population proportions P1−P2P_1 – P_2P1−P2 is (0.0077, 0.1123). This means we are 95% confident that the true difference in proportions between population one and population two lies between 0.0077 and 0.1123.
