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Chapter 3: Problem 86

Construct two sets of numbers with at least five numbers in each set with the following characteristics: The means are the same, but the standard deviation of one of the sets is larger than that of the other. Report the mean and both standard deviations.

Short Answer

Expert verified
The two sets with at least five numbers each, a same mean, and different standards deviations are: {5, 10, 15, 10, 10} and {8, 10, 12, 10, 10}. Both have a mean of 10. The first set has a standard deviation of approximately 3.16 and the second set has a standard deviation of 2.

Step by step solution

01

Constructing two sets with the same mean

Start by constructing two sets with the same mean. For example, the sets {5, 10, 15} and {8, 10, 12} both have a mean of 10. Let's add two more numbers to each set to make the count at least five, like {5, 10, 15, 10, 10} and {8, 10, 12, 10, 10}
02

Compute the standard deviations

Next, compute the standard deviations of both sets. Use the formula for standard deviation, which is the square root of the variance. The variance is the average of the squared differences from the mean. So for the first set, the differences from the mean are -5, 0, 5, 0, 0. Squaring these differences and averaging them gives the variance, 10. The square root of 10 is approximately 3.16. For the second set, the differences from the mean are -2, 0, 2, 0, 0. Squaring these differences gives the variance of 4, and the square root of 4 is 2. Thus, the first set has a larger standard deviation than the second, while both sets have the same mean.
03

Checking the Mean and Standard Deviation

Check that the two sets {5, 10, 15, 10, 10} and {8, 10, 12, 10, 10} have the same mean, 10, and that the first set has a standard deviation of approximately 3.16, while the second set has a standard deviation of 2. You can conclude that your number sets are correct as they both have the same mean but different standard deviations, and the standard deviation of the first set is greater than the second.

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