Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To find the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups, follow these step-by-step calculations:
1. Sample Proportions:
- [tex]\( \hat{p}_1 \)[/tex] is the proportion of voters aged 18-30 who support the increase: [tex]\( \hat{p}_1 = \frac{62}{85} = 0.73 \)[/tex].
- [tex]\( \hat{p}_2 \)[/tex] is the proportion of voters aged 31-40 who support the increase: [tex]\( \hat{p}_2 = \frac{56}{70} = 0.80 \)[/tex].
2. Sample Sizes:
- [tex]\( n_1 = 85 \)[/tex] for voters aged 18-30.
- [tex]\( n_2 = 70 \)[/tex] for voters aged 31-40.
3. Standard Errors of the Sample Proportions:
- Standard error for [tex]\( \hat{p}_1 \)[/tex]: [tex]\( SE_1 = \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} = \frac{0.73 \times (1 - 0.73)}{85} = 0.0028157142857142855 \)[/tex].
- Standard error for [tex]\( \hat{p}_2 \)[/tex]: [tex]\( SE_2 = \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} = \frac{0.80 \times (1 - 0.80)}{70} = 0.0018823529411764702 \)[/tex].
4. Combined Standard Error:
- Combined standard error: [tex]\( SE_{\text{combined}} = \sqrt{SE_1 + SE_2} = \sqrt{0.0028157142857142855 + 0.0018823529411764702} = 0.0685424483578662 \)[/tex].
5. Difference in Sample Proportions:
- [tex]\( \hat{p}_1 - \hat{p}_2 = 0.73 - 0.80 = -0.07000000000000006 \)[/tex].
6. Z-Score for 99% Confidence Interval:
- [tex]\( z = 2.58 \)[/tex].
7. Margin of Error:
- Margin of error: [tex]\( ME = z \times SE_{\text{combined}} = 2.58 \times 0.0685424483578662 = 0.1768395167632948 \)[/tex].
8. Confidence Interval:
- Lower bound: [tex]\( (\hat{p}_1 - \hat{p}_2) - ME = -0.07000000000000006 - 0.1768395167632948 = -0.24683951676329485 \)[/tex].
- Upper bound: [tex]\( (\hat{p}_1 - \hat{p}_2) + ME = -0.07000000000000006 + 0.1768395167632948 = 0.10683951676329473 \)[/tex].
So, the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups is:
[tex]\[ (-0.24683951676329485, 0.10683951676329473) \][/tex]
1. Sample Proportions:
- [tex]\( \hat{p}_1 \)[/tex] is the proportion of voters aged 18-30 who support the increase: [tex]\( \hat{p}_1 = \frac{62}{85} = 0.73 \)[/tex].
- [tex]\( \hat{p}_2 \)[/tex] is the proportion of voters aged 31-40 who support the increase: [tex]\( \hat{p}_2 = \frac{56}{70} = 0.80 \)[/tex].
2. Sample Sizes:
- [tex]\( n_1 = 85 \)[/tex] for voters aged 18-30.
- [tex]\( n_2 = 70 \)[/tex] for voters aged 31-40.
3. Standard Errors of the Sample Proportions:
- Standard error for [tex]\( \hat{p}_1 \)[/tex]: [tex]\( SE_1 = \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} = \frac{0.73 \times (1 - 0.73)}{85} = 0.0028157142857142855 \)[/tex].
- Standard error for [tex]\( \hat{p}_2 \)[/tex]: [tex]\( SE_2 = \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} = \frac{0.80 \times (1 - 0.80)}{70} = 0.0018823529411764702 \)[/tex].
4. Combined Standard Error:
- Combined standard error: [tex]\( SE_{\text{combined}} = \sqrt{SE_1 + SE_2} = \sqrt{0.0028157142857142855 + 0.0018823529411764702} = 0.0685424483578662 \)[/tex].
5. Difference in Sample Proportions:
- [tex]\( \hat{p}_1 - \hat{p}_2 = 0.73 - 0.80 = -0.07000000000000006 \)[/tex].
6. Z-Score for 99% Confidence Interval:
- [tex]\( z = 2.58 \)[/tex].
7. Margin of Error:
- Margin of error: [tex]\( ME = z \times SE_{\text{combined}} = 2.58 \times 0.0685424483578662 = 0.1768395167632948 \)[/tex].
8. Confidence Interval:
- Lower bound: [tex]\( (\hat{p}_1 - \hat{p}_2) - ME = -0.07000000000000006 - 0.1768395167632948 = -0.24683951676329485 \)[/tex].
- Upper bound: [tex]\( (\hat{p}_1 - \hat{p}_2) + ME = -0.07000000000000006 + 0.1768395167632948 = 0.10683951676329473 \)[/tex].
So, the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups is:
[tex]\[ (-0.24683951676329485, 0.10683951676329473) \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.