# Confidence Intervals

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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#### Solution Preview

Please see attached file.

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.

Z-score = (X-mean) / standard deviation.

Thus putting all the values of X, mean and standard deviation we get:

X Z-score

200 (200-250)/30=-1.67

320 (320-250)/30=2.33

220 (220-250)/30=-1

270 (270-250)/30=0.67

250 (250-250)/30=0

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

220 and 280 is within +/- 1 standard deviation of the mean. Thus by empirical rule 68% of the data should be between those numbers.

190 and 310 is within +/- 2 standard deviation of the mean. Thus by empirical rule 95% of the data should be between those numbers.

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

First we need to calculate the z score. That will be (200-250)/30 = -1.67

Now the percentage of data point can be calculated as follows:

1-(2*P[z<-1.67]) =90.5%

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

If we look at the formula, z score measures the number of standard deviation the value is away from the mean. Mean is 0 standard deviation units away from the mean. Thus the z -score of mean is 0.

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?

z score for English Test: (34-28)/5 = 1.2

z score for Math Test: (28-25)/2=1.5

Since z score of Math Test is higher, the student scored higher in the Math test.

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.

Z-score = (X-mean) / standard deviation.

Thus putting all the values of X, mean and standard deviation we ...

#### Solution Summary

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.

Confidence Intervals, Samples, and Unknown Variance

Let (X1, X2,..., Xn) be a sample, where each Xi is a random variable of normal distribution with mean mu and variance sigma². Let us suppose that n = 20, and sigma² = 9. An experiment has yielded the results (X1, X2,..., X20), and we have calculated that the empirical mean x20 = 2.09.

1) Give a confidence interval with level of confidence 90% for mu.

2) How big would the sample have to be for the interval to be half as long?

3) Let us now suppose the variance is not known. Knowing that counting from i=1 to 20 (xi - x20)² = 14.6, give a confidence interval with level of confidence 90% for the value of mu.

4) We now suppose we know mu is known and is equal to 2. Give a confidence interval with a confidence level of 90% for the value of sigma².

5) Same question if we do not know the value of mu.

Please see attachment for proper format.

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