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Statistics for Describing, Exploring, and Comparing Data

Degrees of Freedom: Ten values have a mean of 75.0. Nine of the values are 62,78,90,87,56,92,70,70, and 93.

a. Find the 10th value.
b. We need to create a list of n values that have a specific known mean. We are free to select any values we desire for some of the n values. How many of the n values can be freely assigned before the remaining values are determined? (The result is often referred to as the number of degrees of freedom.)

Range Rule of Thumb: Aluminum cans with a thickness of 0.0111 in. have axial loads with a mean of 281.8 lb and a standard deviation of 27.8 lb. The axial load is measured by applying pressure to the top of the can until it collapses. Use the range rule of thumb to find the minimum and maximum "usual" axial loads. One particular can had an axial load of 504 lb. Is that unusual?

Comparing Scores: Three students take equivalent stress tests. Which is the highest relative score?

a. A score of 144 on a test with a mean of 128 and a standard deviation of 34.
b. A score of 90 on a test with a mean of 86 and a standard deviation of 18.
c. A score of 18 on a test with a mean of 15 and a standard deviation of 5.

Solution Preview

Degrees of Freedom: Ten values have a mean of 75.0. Nine of the values are 62,78,90,87,56,92,70,70, and 93.

a. Find the 10th value.

To find the mean (the average), you add up all the values, then divide the total by the number of values that you have. We can use this to set up an equation and solve for the missing value (we'll call the missing value x).

75.0 = (62 + 78 + 90 + 87 + 56 + 92 + 70 + 70 + x)/10
750 = 62 + 78 + 90 + 87 + 56 + 92 + 70 + 70 + x
750 = 605 + x
145 = x

The missing value is 145. Now that we have all the values, find the mean of the 10 values to verify that it's equal to 75.

b. We need to create a list of n values that have a specific known mean. We are free to select any values we desire for some of the n values. How many of the n values can be freely assigned before the remaining values are determined? (The result is often referred to as the number of degrees of freedom.)

Whenever you have one group of n numbers, the degrees of freedom is equal to n - 1.

In the previous problem, we knew the values of n - 1 numbers (the 9 we were given) and using this information, we were able to find the 10th number. What if we were given n - 2 numbers? Pretend that we ...

Solution Summary

This problem set has three questions concerning means and standard deviations, degrees of freedom, range of a data set, and relative scores. The solution provides answers and explanations to all parts of the problem set.

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