A professor records the time (in minutes) that it takes 16 students to complete an exam. Compute the SS, the variance, and the standard deviation assuming the 16 students constitute a population and assuming the 16 students constitute a sample. (Round your answers for variance and standard deviation to two decimal places.)
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Why didn't you follow directions?
To compute the SS (sum of squares), variance, and standard deviation, we need to follow certain steps:
1. Find the mean (average) of the recorded times for the 16 students.
2. Compute the deviation of each recorded time from the mean by subtracting the mean from each time. Squaring each deviation will give us squared deviations.
3. Add up all the squared deviations to get the sum of squares (SS).
4. Calculate the variance by dividing the SS by the number of students (16 if they constitute a population) or by (n-1) if they constitute a sample, where n is the number of students.
5. Finally, take the square root of the variance to get the standard deviation.
Let's go through the steps:
1. Find the mean:
To find the mean, sum up all the recorded times and divide by the total number of students (16 in this case).
2. Compute the deviation:
Subtract the mean from each recorded time to get the deviation.
3. Squared deviations:
Square each deviation to get the squared deviations.
4. Sum of squares (SS):
Add up all the squared deviations to get the SS.
5. Variance:
If the 16 students constitute a population, divide the SS by 16 to get the variance. If they are a sample, divide the SS by (16-1), which is 15, to get the variance.
6. Standard deviation:
Take the square root of the variance to get the standard deviation.
Now let's calculate the values:
Step 1: Find the mean:
Add up all the recorded times and divide by 16 to get the mean.
Step 2: Compute the deviation:
Subtract the mean from each recorded time.
Step 3: Squared deviations:
Square each deviation.
Step 4: Sum of squares (SS):
Add up all the squared deviations.
Step 5: Variance:
If the 16 students constitute a population, divide the SS by 16 for the variance. If they are a sample, divide the SS by 15.
Step 6: Standard deviation:
Take the square root of the variance.
Finally, round the variance and standard deviation to two decimal places.