a) Banting is believed to reduce weight when done for a minimum of 10 days. A random sample of 5 ladies were weighed before and after Banting. The data is displayed on a table below. Person Weight before Weight After 85 96 105 78 90 120 Construct a 95% for the difference in mean weight. [8 Marks] b) Suppose you want to know the internet speed of two different mobile networks. Network X Network Y Sample size Sample Mean Sample std dev 60 50 Assuming that the population is normally distributed with equal variances Construct a 95% for the difference in mean internet
Sure! Let's break down the process step by step:
a) To construct a confidence interval for the difference in mean weight, we can use the formula:
CI = X̄ ± t * (s / √n)
1. Calculate the mean difference in weight:
- Subtract each person's weight before from their weight after to find the difference in weight for each person.
- Then, find the mean of these differences.
2. Calculate the standard deviation of the differences:
- Subtract each person's difference in weight from the mean difference.
- Square each result, sum them up, and divide by (n-1) where n is the sample size.
- Finally, take the square root of this value.
3. Determine the critical value:
- Look up the critical value for a 95% confidence interval using a t-distribution table with (n-1) degrees of freedom. In this case, since you have a sample size of 5, the degrees of freedom is (5-1) = 4.
4. Calculate the standard error:
- Divide the standard deviation of the differences by the square root of the sample size.
5. Calculate the margin of error:
- Multiply the critical value by the standard error.
6. Construct the confidence interval:
- Calculate the upper and lower bounds of the confidence interval by adding and subtracting the margin of error from the mean difference in weight.
b) To construct a confidence interval for the difference in mean internet speed, we can follow similar steps:
1. Calculate the mean difference in internet speed:
- Subtract the sample mean of network Y from the sample mean of network X.
2. Calculate the standard deviation of the differences:
- Calculate the pooled estimate of the variance by combining the sample standard deviations of both networks.
3. Determine the critical value:
- Look up the critical value for a 95% confidence interval using a t-distribution table with (nX + nY - 2) degrees of freedom. In this case, nX = 60 and nY = 50.
4. Calculate the standard error:
- Divide the pooled estimate of the variance by the square root of the sum of the sample sizes.
5. Calculate the margin of error:
- Multiply the critical value by the standard error.
6. Construct the confidence interval:
- Calculate the upper and lower bounds of the confidence interval by adding and subtracting the margin of error from the mean difference in internet speed.
Remember to use the appropriate t-distribution table and degrees of freedom for each scenario when determining the critical value.
I hope this explanation helps you construct the confidence intervals!