Find the 98% confidence interval for estimating ìd based on these paired data and assuming normality. (Give your answers correct to one decimal place.)
Before 50 55 61 52 68 54
After 34 58 56 36 40 39
Lower Limit Incorrect: Your answer is incorrect. 64.0
Upper Limit Incorrect: Your answer is incorrect. 125.4
I don't know how I missed them? I had mean 87.75, sd 30.27, vari 16969.29545
To find the 98% confidence interval for estimating μd (the population mean difference) based on the given paired data, we can follow these steps:
1. Calculate the differences between the "Before" and "After" values to get the paired differences:
Before: 50, 55, 61, 52, 68, 54
After: 34, 58, 56, 36, 40, 39
Paired Differences: 34-50, 58-55, 56-61, 36-52, 40-68, 39-54
= -16, 3, -5, -16, -28, -15
= -16, 3, -5, -16, -28, -15
2. Calculate the sample mean of the paired differences:
Mean (x̄d) = (Σd) / n
where Σd is the sum of the paired differences and n is the sample size.
Σd = -16 + 3 + -5 + -16 + -28 + -15
= -77
Mean (x̄d) = -77 / 6
≈ -12.83 (rounded to two decimal places)
3. Calculate the sample standard deviation of the paired differences:
Standard Deviation (s) = sqrt((Σd^2 - (Σd)^2 / n) / (n - 1))
where Σd^2 is the sum of the squared paired differences.
Σd^2 = (-16)^2 + 3^2 + (-5)^2 + (-16)^2 + (-28)^2 + (-15)^2
= 256 + 9 + 25 + 256 + 784 + 225
= 1555
Standard Deviation (s) = sqrt((1555 - (-77)^2 / 6) / (6 - 1))
= sqrt((1555 - 5929 / 6) / 5)
= sqrt((1555 - 1054.67) / 5)
= sqrt(500.33 / 5)
= sqrt(100.07)
≈ 10.00 (rounded to two decimal places)
4. Calculate the standard error:
Standard Error (SE) = s / sqrt(n)
where n is the sample size.
SE = 10.00 / sqrt(6)
≈ 4.082 (rounded to three decimal places)
5. Calculate the margin of error based on the 98% confidence level:
Margin of Error = critical value * SE
The critical value for a 98% confidence level can be found using a t-distribution with n - 1 degrees of freedom. Since n = 6, the degrees of freedom = 6 - 1 = 5.
Using a t-table or calculator, the critical value for a 98% confidence level and 5 degrees of freedom is approximately 2.571.
Margin of Error = 2.571 * 4.082
≈ 10.488 (rounded to three decimal places)
6. Calculate the lower and upper limits of the confidence interval:
Lower Limit = x̄d - Margin of Error
= -12.83 - 10.488
≈ -23.318 (rounded to three decimal places)
Upper Limit = x̄d + Margin of Error
= -12.83 + 10.488
≈ -2.342 (rounded to three decimal places)
Therefore, the 98% confidence interval for estimating μd is approximately (-23.3, -2.3) (rounded to one decimal place).
Note: It seems that there was an error in the calculations or rounding when finding the confidence interval. Please double-check your calculations and make sure to use the correct formulas to obtain the accurate interval.