# This problem calculates z-scores and confidence intervals (using Chebyshev inequality). It also explains why the z-score of the mean needs to be 0. A number of examples have been provided.

1. Consider an infinite population with a normal shape and a mean of 250 and standard deviation of 30.

a. Compute the z-scores for the following values of X and locate each on the graph.

X Z-score
200
320
220
270
250

b. According to the Empirical rule what percent of the data should be between 220 and 280? Between 190 and 310?

c. According to Chebyshev what percent should be between 200 and 300

d. Why is the z-score of the mean zero?

e. A student scores 34 on and English test that has a mean of 28 and a standard deviation of 5. He scores a 28 on a math test that has a mean of 25 and a standard deviation of 2. Which score is higher and why?

Group 2:
2. Consider an infinite population with a normal shape and a mean of 500 and standard deviation of 100.

a. Compute the z-scores for the following values of X and locate each on the graph.

X Z-score
800
350
620
500
250

b. According to the Empirical rule what percent of the data should be between 400 and 600? Between 300 and 700?

c. According to Chebyshev what percent should be between 250 and 750

d. Why is the z-score of the mean zero?

e. A student scores 31 on and English test that has a mean of 28 and a standard deviation of 5. He scores a 28 on a math test that has a mean of 25 and a standard deviation of 7. Which score is higher and why?

Group 3

3. Consider an infinite population with a normal shape and a mean of 80 and standard deviation of 16.

a. Compute the z-scores for the following values of X and locate each on the graph.

X Z-score
100
56
80
72
85

b. According to the Empirical rule what percent of the data should be between 64 and 96? Between 48 and 112?

c. According to Chebyshev what percent should be between 56 and 104

d. Why is the z-score of the mean zero?

e. A student scores 36 on and English test that has a mean of 28 and a standard deviation of 5. He scores a 29 on a math test that has a mean of 25 and a standard deviation of 2. Which score is higher and why?

Group 4

4. Consider an infinite population with a normal shape and a mean of 250 and standard deviation of 60.

a. Compute the z-scores for the following values of X and locate each on the graph.

X Z-score
100
320
420
250
190

b. According to the Empirical rule what percent of the data should be between 190 and 310? Between 130 and 370?

c. According to Chebyshev what percent should be between 130 and 370

d. Why is the z-score of the mean zero?

e. A student scores 34 on and English test that has a mean of 28 and a standard deviation of 10. He scores a 28 on a math test that has a mean of 25 and a standard deviation of 3. Which score is higher and why?

Group 5
5. Consider an infinite population with a normal shape and a mean of 300 and standard deviation of 30.
a. Compute the z-scores for the following values of X and locate each on the graph.

X Z-score
200
360
220
270
300

b. According to the Empirical rule what percent of the data should be between 270 and 330? Between 240 and 360?

c. According to Chebyshev what percent should be between 240 and 360

d. Why is the z-score of the mean zero?

e. A student scores 33 on and English test that has a mean of 28 and a standard deviation of 5. He scores a 27 on a math test that has a mean of 25 and a standard deviation of 2. Which score is higher and why?

See attached file for full problem description.

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