2. [4 points] Serum cholesterol levels were monitored to investigate the effect of physical training. Prior to this training, blood samples were acquired from each of the 12 subjects and were tested for cholesterol level. After a week-long training period of daily running, the same individuals were retested, yielding the following data:
PRE-TRAINING
182
232
191
200
148
249
276
213
241
480
262
189
SUBJECT
POST-TRAINING
A 198
B 210
C 194
D 220
E 138
F 220
G 219
H 161
I 210
J 313
K 226
L 202
Compute a 90% confidence interval for the mean decrease of cholesterol level after physical training. You do not need to calculate any standard deviations by hand—feel free to use a calculator or spreadsheet for this.
[Be careful: you are being asked for the mean decrease, not the decrease of the mean! You may assume n-1 degrees of freedom, and will need to estimate a value of t from your table.]
To compute a confidence interval for the mean decrease of cholesterol level after physical training, we need to follow these steps:
Step 1: Calculate the difference between the pre-training and post-training cholesterol levels for each subject.
To calculate the difference, subtract the post-training cholesterol level from the pre-training cholesterol level for each subject. For example, for subject A, the difference is: 182 - 198 = -16.
Here is the table with the differences:
SUBJECT DIFFERENCE
A -16
B -10
C -3
D -20
E 10
F 29
G 57
H 52
I -31
J -167
K -36
L -13
Step 2: Calculate the mean and standard deviation of the differences.
Calculate the mean of the differences by summing all the differences and dividing by the number of subjects (n). In this case, n = 12.
Mean = (-16 - 10 - 3 - 20 + 10 + 29 + 57 + 52 - 31 - 167 - 36 - 13) / 12 = 2.75
Calculate the standard deviation of the differences using the sample formula (since we have a small sample size).
Standard deviation = sqrt((sum of (difference - mean)^2) / (n - 1))
Using the formula, we get:
Standard deviation = sqrt(((-16 - 2.75)^2 + (-10 - 2.75)^2 + ... + (-13 - 2.75)^2) / (12 - 1))
This calculation can be performed using a calculator or spreadsheet software.
Step 3: Determine the critical value (t-value) for a 90% confidence interval.
To determine the critical value, we need to estimate a t-value from a t-distribution table. Based on a 90% confidence level and n-1 degrees of freedom (n-1 = 12 - 1 = 11), the t-value is approximately 1.796. You can find this value in a t-distribution table or use statistical software.
Step 4: Calculate the margin of error.
The margin of error is the product of the critical value and the standard deviation divided by the square root of the sample size.
Margin of error = (t-value * standard deviation) / sqrt(n)
In this case, margin of error = (1.796 * calculated standard deviation) / sqrt(12)
Step 5: Calculate the confidence interval.
The confidence interval is calculated by subtracting the margin of error from the mean and adding the margin of error to the mean.
Lower bound = mean - margin of error
Upper bound = mean + margin of error
In this case, lower bound = 2.75 - margin of error and upper bound = 2.75 + margin of error.
Using the given information and calculated values, you can plug in the numbers to compute the 90% confidence interval for the mean decrease of cholesterol level after physical training.