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# Chebyshev's Theorem, Empirical Rule and Descriptive Stats

1. a) Use Chebyshev's theorem to find what percent of the values will fall between 220 and 316 for a data set with a mean of 268 and standard deviation of 12.

b) Use the Empirical Rule to find what two values 99.7% of the data will fall between for a data set with a mean of 239 and standard deviation of 19.

2. A Math test has a mean of 45 and standard deviation of 10.0. Find the corresponding z scores for:

a) a test score of 13

b) a test score of 54

3. Rank the following data in increasing order and find the position and value of the 91st percentile.
4 5 4 9 9 1 0 7 4 8 0 9

4. The following data lists the average monthly snowfall for January in 15 cities around the US: Find the mean, variance, and standard deviation.

43 16 38 28 7 31 18 37
35 5 43 23 24 29 22

5. Starting with the data values 70 and 100, add three data values to the sample so that the mean is 85, the median is 76, and the mode is 76.

#### Solution Preview

1. a) Chebyshev's theorem states that 1 - (1/k^2) of the values will fall between Mean-(k*SD), Mean+(k*SD).

In this instance the k value is 4.0 as each of the values are 4 standard deviations away from the mean.

1 - (1/4^2) = 0.9375

Therefore 93.75% of the values fall between (220, 316).

b) The empirical rule states that approximately 99.7% of values will fall within three standard deviations of the mean, so then the interval (182, 296) will actually contain 99.7% of observed data. ...

#### Solution Summary

Complete worked through solutions for several types of statistics problems. Included are example problems for Chebyshev's theorem, the Empirical Rule, z-scores, mean, variance, standard deviation, and percentile calculations.

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