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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]
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