A test of writing ability is given to a random sample of students before and after they completed a formal writing course. The results are given below. Construct a 99% confidence interval for the mean difference between the before and after scores. You may assume the populations are normally distributed.
Before 70 80 92 99 93 97 76 63 68 71 74
After 69 79 90 96 91 95 75 64 62 64 76
To construct a confidence interval for the mean difference between the before and after scores, we need to follow these steps:
Step 1: Calculate the difference between the before and after scores for each student.
Before: 70 80 92 99 93 97 76 63 68 71 74
After: 69 79 90 96 91 95 75 64 62 64 76
Difference: 1 1 2 3 2 2 1 -1 6 7 2
Step 2: Find the mean and standard deviation of the differences.
Mean (x̄) = Σdifference / n = (1+1+2+3+2+2+1-1+6+7+2) / 11 = 24 / 11 = 2.18 (rounded to two decimal places)
Standard Deviation (s) = √[(Σ(difference - mean)^2) / (n - 1)] = √[(1.42 + 1.42 + 0.82 + 0.02 + 0.82 + 0.82 + 1.42 + 13.72 + 18.42 + 35.52 + 0.02) / 10] = √(77.52 / 10) = √7.75 = 2.78 (rounded to two decimal places)
Step 3: Determine the critical value.
Since we want to construct a 99% confidence interval, we need to find the critical value corresponding to a 99% confidence level and a two-tailed test. For this, we can use a t-distribution and find the t-value with 10 degrees of freedom (n - 1).
Using a t-distribution table or calculator, the critical value for a 99% confidence level with 10 degrees of freedom is approximately 3.169.
Step 4: Calculate the margin of error.
The margin of error is given by the formula: margin of error = critical value * standard deviation / √n.
Margin of Error = 3.169 * 2.78 / √11 ≈ 3.097
Step 5: Construct the confidence interval.
The confidence interval can now be constructed using the formula: CI = mean ± margin of error.
CI = 2.18 ± 3.097
Lower bound = 2.18 - 3.097 = -0.917 (rounded to three decimal places)
Upper bound = 2.18 + 3.097 = 5.277 (rounded to three decimal places)
Therefore, the 99% confidence interval for the mean difference between the before and after scores is approximately -0.917 to 5.277.