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 nonsense words. Each was asked to list as many of the words as they
could remember both 1 hour and 24 hours later. The data are given in the accompanying table.
Subject 1 2 3 4 5 6 7 8
1 hr later 14 12 18 7 11 9 16 15
24 hr later 10 4 13 6 9 6 13 12
Construct a 90% confidence interval for the difference in mean number of words remembered after 1 hour
and after 24 hours. (Use 1 Hour Later − 24 Hours Later. Enter your answer using interval notation. Round
your numerical values to four decimal places.)
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To construct a confidence interval for the difference in mean number of words remembered after 1 hour and after 24 hours, we can use the following formula:
Confidence Interval = (Mean Difference) ± (Critical Value) * (Standard Error)
First, we need to calculate the mean difference between the two time periods:
Mean Difference = (1 Hour Later) - (24 Hours Later)
Mean Difference = (14+12+18+7+11+9+16+15) - (10+4+13+6+9+6+13+12)
Mean Difference = 38 - 73
Mean Difference = -35
Next, we need to calculate the standard error:
Standard Error = (Standard Deviation) / √(Sample Size)
To find the standard deviation, we need to calculate the sample variances for each time period:
Sample Variance (1 Hour Later) = [(14-38)^2 + (12-38)^2 + (18-38)^2 + (7-38)^2 + (11-38)^2 + (9-38)^2 + (16-38)^2 + (15-38)^2] / (8-1)
Sample Variance (1 Hour Later) = 2032 / 7
Sample Variance (1 Hour Later) = 290.2857
Sample Variance (24 Hours Later) = [(10-73)^2 + (4-73)^2 + (13-73)^2 + (6-73)^2 + (9-73)^2 + (6-73)^2 + (13-73)^2 + (12-73)^2] / (8-1)
Sample Variance (24 Hours Later) = 25366 / 7
Sample Variance (24 Hours Later) = 3623.7143
Now we can find the standard error:
Standard Error = √[(Sample Variance 1 Hour Later) / (Sample Size 1 Hour Later) + (Sample Variance 24 Hours Later) / (Sample Size 24 Hours Later)]
Standard Error = √[290.2857 / 8 + 3623.7143 / 8]
Standard Error = √(36.2857 + 452.9643)
Standard Error = √489.25
Standard Error = 22.125
Finally, we need to determine the critical value for a 90% confidence interval. Since we have a small sample size (n < 30) and the population standard deviation is unknown, we can use the t-distribution. For a 90% confidence interval with 7 degrees of freedom, the critical value is approximately 1.895.
Now we can calculate the confidence interval:
Confidence Interval = (-35) ± (1.895) * (22.125)
Confidence Interval = -35 ± 41.890
Confidence Interval = (-76.890, 6.890)
Therefore, the 90% confidence interval for the difference in mean number of words remembered after 1 hour and after 24 hours is (-76.890, 6.890).
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