Nine students took the SAT. Their scores are listed below. Later on, they read a book on test preparation and retook the SAT. Their new scores are listed below. Construct a 95% confidence interval for µ d (the true mean difference in scores). Assume that the distribution is normally distributed.
Where are the scores
To construct a 95% confidence interval for the mean difference in scores (µd), we can follow these steps:
1. Calculate the difference in scores for each student by subtracting their initial SAT score from their retake SAT score.
2. Calculate the sample mean of the differences (x̄d) by summing up all the differences and dividing by the number of students.
3. Calculate the sample standard deviation of the differences (s𝑑) using the formula:
s𝑑 = √[Σ(𝑥𝑑 - 𝑥̄𝑑)² / (n - 1)]
where Σ represents the sum of squares, 𝑥𝑑 is each individual difference, 𝑥̄𝑑 is the sample mean difference, and n is the number of students.
4. Calculate the standard error (SE) of the mean difference using the formula:
SE = s𝑑 / √n
5. Determine the critical value corresponding to the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.
6. Calculate the margin of error (ME) by multiplying the standard error by the critical value:
ME = SE × critical value
7. Finally, construct the confidence interval by subtracting the margin of error from the sample mean difference and adding the margin of error to the sample mean difference:
CI = x̄d - ME, x̄d + ME
Let's calculate the confidence interval step by step. Please provide the dataset of initial and retake SAT scores for the nine students.
To construct a 95% confidence interval for the true mean difference in scores (µd), we need to follow these steps:
1. Calculate the differences between the initial (before reading the book) and final (after reading the book) SAT scores for each student.
2. Find the mean (x̄d) and standard deviation (s) of the differences.
3. Determine the critical value (z*) for a 95% confidence level. This value can be found using a standard normal distribution table or a calculator.
4. Calculate the margin of error (E), which is equal to z* multiplied by the standard deviation of the differences (s) divided by the square root of the sample size (n).
5. Finally, construct the confidence interval by subtracting the margin of error (E) from the mean difference (x̄d) and adding the margin of error (E) to the mean difference (x̄d).
Let's go through each step in detail:
Step 1: Calculate the differences between the initial and final SAT scores for each student:
Let's say the initial scores are denoted as X1, X2, X3, ..., X9, and the final scores are Y1, Y2, Y3, ..., Y9. Calculate the differences by subtracting the initial scores from the final scores: D1 = Y1 - X1, D2 = Y2 - X2, ..., D9 = Y9 - X9.
Step 2: Find the mean (x̄d) and standard deviation (s) of the differences:
Calculate the mean difference (x̄d) by summing up all the differences (D1, D2, ..., D9) and dividing by the sample size (n = 9): x̄d = (D1 + D2 + ... + D9) / 9.
Calculate the standard deviation (s) of the differences using the formula:
s = sqrt[(Σ(Di - x̄d)^2) / (n - 1)], where Σ represents the sum, Di is each difference, and x̄d is the mean difference.
Step 3: Determine the critical value (z*) for a 95% confidence level:
Since the distribution is assumed to be normally distributed, you can use a standard normal distribution table or a calculator to find the critical value (z*) associated with a 95% confidence level. For a two-tailed test at a 95% confidence level, z* is approximately 1.96.
Step 4: Calculate the margin of error (E):
The margin of error (E) is calculated using the formula:
E = z* * (s / sqrt(n)), where z* is the critical value, s is the standard deviation, and sqrt(n) is the square root of the sample size.
Step 5: Construct the confidence interval:
The confidence interval is calculated by subtracting the margin of error (E) from the mean difference (x̄d) and adding the margin of error (E) to the mean difference (x̄d):
Confidence interval = (x̄d - E, x̄d + E)
That's it! By following these steps, you can construct a 95% confidence interval for the true mean difference in scores (µd).
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
95% = mean ± 1.96 SEm
SEm = SD/√n
You can do the calculations.