In a study of memory recall, 8 students from a large psychology class were selected at random and given
10 minutes to memorize a list of 20 nonsensical words. Each was asked to list as many of the words as he
or she could remember both 1 hour and 24 hours later. The data are as shown in the accompanying table.
Student 1 2 3 4 5 6 7 8
1 hour later 14 12 20 7 12 10 18 15
24 hours later 9 3 15 5 9 6 13 11
Use these data to estimate the difference in mean number of words remembered after 1 hour and after 24
hours. Use a 90% confidence interval
Use 𝜇d = 𝜇1hourlater − 𝜇24hourslater
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To estimate the difference in mean number of words remembered after 1 hour and after 24 hours, we first calculate the difference for each student. Then we can calculate the mean difference and the standard error of the mean difference.
1. Calculate the difference for each student:
- Student 1: 14 - 9 = 5
- Student 2: 12 - 3 = 9
- Student 3: 20 - 15 = 5
- Student 4: 7 - 5 = 2
- Student 5: 12 - 9 = 3
- Student 6: 10 - 6 = 4
- Student 7: 18 - 13 = 5
- Student 8: 15 - 11 = 4
2. Calculate the mean difference:
Mean difference = (5 + 9 + 5 + 2 + 3 + 4 + 5 + 4) / 8 = 37 / 8 = 4.625
3. Calculate the standard deviation of the mean difference:
Calculate the variance: Var(d) = Σ(d^2) / (n-1), where d is the difference for each student and n is the number of students.
- (5-4.625)^2 + (9-4.625)^2 + (5-4.625)^2 + (2-4.625)^2 + (3-4.625)^2 + (4-4.625)^2 + (5-4.625)^2 + (4-4.625)^2 = 20.359375
- Var(d) = 20.359375 / (8-1) = 2.765625
Calculate the standard error of the mean difference: SE(d) = sqrt(Var(d) / n) = sqrt(2.765625 / 8) = 0.59012855
4. Calculate the margin of error:
Margin of error = t * SE(d), where t is the t critical value for a 90% confidence interval with n-1 degrees of freedom.
For 8 degrees of freedom, the t critical value at a 90% confidence interval is approximately 1.860
Margin of error = 1.860 * 0.59012855 = 1.097639
5. Calculate the confidence interval:
Confidence interval = (Mean difference - Margin of error, Mean difference + Margin of error)
Confidence interval = (4.625 - 1.097639, 4.625 + 1.097639) = (3.527361, 5.722639)
Therefore, we can estimate that the difference in mean number of words remembered after 1 hour and after 24 hours is between 3.527361 and 5.722639 with 90% confidence.
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