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 each pair of before and after scores:
Difference = After - Before
Difference = 69-70, 79-80, 90-92, 96-99, 91-93, 95-97, 75-76, 64-63, 62-68, 64-71, 76-74
Step 2: Calculate the mean and the standard deviation of the differences:
Mean (X̄) = Mean(Difference)
Standard Deviation (s) = Standard Deviation(Difference)
Step 3: Determine the sample size (n) or the number of pairs in the sample.
Step 4: Calculate the standard error (SE), which is the standard deviation divided by the square root of the sample size:
SE = s / √n
Step 5: Determine the critical value (z*) for a 99% confidence level. To find this value, we will use the Z-table or a calculator. At the 99% confidence level, the critical value is approximately 2.896.
Step 6: Calculate the margin of error (ME) using the following formula:
ME = z* * SE
Step 7: Determine the lower and upper bounds of the confidence interval:
Lower Bound = Mean - ME
Upper Bound = Mean + ME
Let's now calculate the confidence interval using these steps:
Step 1: Calculating the differences:
Difference = -1, -1, -2, -3, -2, -2, -1, 1, -6, -7, 2
Step 2: Calculate the mean and standard deviation of the differences:
Mean (X̄) = (-1 - 1 - 2 - 3 - 2 - 2 - 1 + 1 - 6 - 7 + 2) / 11 = -1.545
Standard Deviation (s) = sqrt(((-1 - (-1.545))^2 + (-1 - (-1.545))^2 + ... + (2 - (-1.545))^2) / (11 - 1)) = 2.93
Step 3: Determine the sample size (n):
n = 11
Step 4: Calculate the standard error (SE):
SE = s / √n = 2.93 / sqrt(11) = 0.88
Step 5: Determine the critical value (z*) for a 99% confidence level:
z* = 2.896
Step 6: Calculate the margin of error (ME):
ME = z* * SE = 2.896 * 0.88 = 2.55
Step 7: Determine the lower and upper bounds of the confidence interval:
Lower Bound = Mean - ME = -1.545 - 2.55 = -4.095
Upper Bound = Mean + ME = -1.545 + 2.55 = 1.005
Therefore, the 99% confidence interval for the mean difference between the before and after scores is approximately -4.095 to 1.005.
To construct a confidence interval for the mean difference between the before and after scores, you need to follow these steps:
Step 1: Calculate the difference between the before and after scores for each individual.
- 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: 70-69, 80-79, 92-90, 99-96, 93-91, 97-95, 76-75, 63-64, 68-62, 71-64, 74-76
Step 2: Calculate the sample mean difference (d̄) and the sample standard deviation of the differences (s).
- Sample mean difference (d̄): Add up all the differences and divide by the number of differences.
- Sample standard deviation of the differences (s): Calculate the square root of the sum of squared differences divided by (n-1), where n is the number of differences.
Step 3: Calculate the standard error of the mean difference (SE).
- Standard error (SE) = s / sqrt(n), where n is the number of differences.
Step 4: Determine the t-value for a 99% confidence interval using (n-1) degrees of freedom.
- Degrees of freedom (df) = n - 1, where n is the number of differences.
Step 5: Calculate the margin of error (ME).
- Margin of error (ME) = t-value * SE
Step 6: Calculate the lower and upper bounds of the confidence interval.
- Lower bound = d̄ - ME
- Upper bound = d̄ + ME
Now, let's go ahead and calculate the confidence interval:
Step 1: Calculate the difference between the before and after scores for each individual.
The differences are: -1, 1, 2, 3, 2, 2, 1, -1, 6, 7, -2
Step 2: Calculate the sample mean difference (d̄) and the sample standard deviation of the differences (s).
d̄ = (sum of differences) / (number of differences)
= (-1 + 1 + 2 + 3 + 2 + 2 + 1 - 1 + 6 + 7 - 2) / 11
= 24 / 11
≈ 2.18
s = sqrt((sum of squared differences) / (n-1))
= sqrt((-1)^2 + 1^2 + 2^2 + 3^2 + 2^2 + 2^2 + 1^2 + (-1)^2 + 6^2 + 7^2 + (-2)^2)/(11-1))
= sqrt(105/10)
≈ sqrt(10.5)
≈ 3.24
Step 3: Calculate the standard error of the mean difference (SE).
SE = s / sqrt(n)
= 3.24 / sqrt(11)
≈ 0.976
Step 4: Determine the t-value for a 99% confidence interval using (n-1) degrees of freedom.
df = n - 1
= 11 - 1
= 10
We can consult a t-distribution table or use statistical software to find the t-value for a 99% confidence interval with 10 degrees of freedom. The t-value is approximately 3.169.
Step 5: Calculate the margin of error (ME).
ME = t-value * SE
≈ 3.169 * 0.976
≈ 3.092
Step 6: Calculate the lower and upper bounds of the confidence interval.
Lower bound = d̄ - ME
= 2.18 - 3.092
≈ -0.912
Upper bound = d̄ + ME
= 2.18 + 3.092
≈ 5.272
Therefore, the 99% confidence interval for the mean difference between the before and after scores is approximately -0.912 to 5.272.