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Sagot :
To construct a 99% confidence interval for the mean difference in the time it takes to experience relief using treatments A and B, we'll follow these steps:
1. Calculate the mean of the differences:
The differences recorded from the experiment are:
[tex]\(-9, -5, -6, -6, -7, -4, -2, -6, -5, -10\)[/tex].
The mean ([tex]\(\bar{d}\)[/tex]) is the sum of these differences divided by the number of differences (n = 10):
[tex]\[ \bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i = \frac{-9 + (-5) + (-6) + (-6) + (-7) + (-4) + (-2) + (-6) + (-5) + (-10)}{10} = \frac{-60}{10} = -6 \][/tex]
2. Calculate the standard deviation of the differences:
To find the sample standard deviation ([tex]\(s\)[/tex]), you use the formula:
[tex]\[ s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (d_i - \bar{d})^2} \][/tex]
For our data, the value of the standard deviation is:
[tex]\[ s \approx 2.309 \][/tex]
3. Determine the critical value for the 99% confidence interval:
With a 99% confidence interval, the significance level ([tex]\(\alpha\)[/tex]) is 0.01. Since we're interested in a two-tailed test, we split [tex]\(\alpha\)[/tex] into two: [tex]\(\alpha/2 = 0.005\)[/tex]. Given that our sample size is [tex]\(n = 10\)[/tex], the degrees of freedom (df) are [tex]\(n - 1 = 9\)[/tex].
The critical value ([tex]\(t^\ast\)[/tex]) can be found using a t-distribution table or statistical software for [tex]\(df = 9\)[/tex] and [tex]\(\alpha/2 = 0.005\)[/tex]:
[tex]\[ t^\ast \approx 3.250 \][/tex]
4. Calculate the standard error of the mean difference:
The standard error of the mean difference ([tex]\(\text{SE}_\bar{d}\)[/tex]) is calculated as:
[tex]\[ \text{SE}_\bar{d} = \frac{s}{\sqrt{n}} = \frac{2.309}{\sqrt{10}} \approx 0.730 \][/tex]
5. Calculate the margin of error:
The margin of error (MOE) is calculated by multiplying the critical value by the standard error:
[tex]\[ \text{MOE} = t^\ast \times \text{SE}_\bar{d} = 3.250 \times 0.730 \approx 2.373 \][/tex]
6. Construct the confidence interval:
Finally, the 99% confidence interval for the mean difference is constructed by adding and subtracting the margin of error from the sample mean:
[tex]\[ \bar{d} \pm \text{MOE} = -6 \pm 2.373 \][/tex]
So, the confidence interval is:
[tex]\[ (-8.373, -3.627) \][/tex]
### Conclusion
Based on our calculations, the 99% confidence interval for the mean difference in relief time between treatments A and B is approximately [tex]\( (-8.373, -3.627) \)[/tex]. This interval suggests that, on average, treatment A provides relief between approximately 3.63 and 8.37 minutes faster than treatment B for the volunteers in this study.
1. Calculate the mean of the differences:
The differences recorded from the experiment are:
[tex]\(-9, -5, -6, -6, -7, -4, -2, -6, -5, -10\)[/tex].
The mean ([tex]\(\bar{d}\)[/tex]) is the sum of these differences divided by the number of differences (n = 10):
[tex]\[ \bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i = \frac{-9 + (-5) + (-6) + (-6) + (-7) + (-4) + (-2) + (-6) + (-5) + (-10)}{10} = \frac{-60}{10} = -6 \][/tex]
2. Calculate the standard deviation of the differences:
To find the sample standard deviation ([tex]\(s\)[/tex]), you use the formula:
[tex]\[ s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (d_i - \bar{d})^2} \][/tex]
For our data, the value of the standard deviation is:
[tex]\[ s \approx 2.309 \][/tex]
3. Determine the critical value for the 99% confidence interval:
With a 99% confidence interval, the significance level ([tex]\(\alpha\)[/tex]) is 0.01. Since we're interested in a two-tailed test, we split [tex]\(\alpha\)[/tex] into two: [tex]\(\alpha/2 = 0.005\)[/tex]. Given that our sample size is [tex]\(n = 10\)[/tex], the degrees of freedom (df) are [tex]\(n - 1 = 9\)[/tex].
The critical value ([tex]\(t^\ast\)[/tex]) can be found using a t-distribution table or statistical software for [tex]\(df = 9\)[/tex] and [tex]\(\alpha/2 = 0.005\)[/tex]:
[tex]\[ t^\ast \approx 3.250 \][/tex]
4. Calculate the standard error of the mean difference:
The standard error of the mean difference ([tex]\(\text{SE}_\bar{d}\)[/tex]) is calculated as:
[tex]\[ \text{SE}_\bar{d} = \frac{s}{\sqrt{n}} = \frac{2.309}{\sqrt{10}} \approx 0.730 \][/tex]
5. Calculate the margin of error:
The margin of error (MOE) is calculated by multiplying the critical value by the standard error:
[tex]\[ \text{MOE} = t^\ast \times \text{SE}_\bar{d} = 3.250 \times 0.730 \approx 2.373 \][/tex]
6. Construct the confidence interval:
Finally, the 99% confidence interval for the mean difference is constructed by adding and subtracting the margin of error from the sample mean:
[tex]\[ \bar{d} \pm \text{MOE} = -6 \pm 2.373 \][/tex]
So, the confidence interval is:
[tex]\[ (-8.373, -3.627) \][/tex]
### Conclusion
Based on our calculations, the 99% confidence interval for the mean difference in relief time between treatments A and B is approximately [tex]\( (-8.373, -3.627) \)[/tex]. This interval suggests that, on average, treatment A provides relief between approximately 3.63 and 8.37 minutes faster than treatment B for the volunteers in this study.
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