# Chebyshev's Theorem for a Sample of Scores on a Psychology Exam

For a particular sample of 77 scores on a psychology exam, the following results were obtained.

First quartile = 44 Third quartile = 71 Standard deviation = 6 Range = 45

Mean = 64 Median = 57 Mode = 92 Midrange = 60

I. What score was earned by more students than any other score? Why?

II. What was the highest score earned on the exam?

III. What was the lowest score earned on the exam?

IV. According to Chebyshev's Theorem, how many students scored between 40 and 88?

V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 46 and 82?

Please show all of your work.

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#### Solution Preview

(a) 92 was the score obtained by students more than any other score. This is because mode is 92.

(b) Let the highest and lowest scores be H and L respectively. We have H ...

#### Solution Summary

This solution explains the mode, how to solve for highest and lowest score using the midrange and range, the number of students that scored between 40 and 88 according to Chebyshev's theorem and the number of students that scored between 46 and 82 according to the Empirical Rule.

Statistics: distribution, correlation coefficient, variables, regression line

1. For a particular sample of 50 scores on a psychology exam, the following results were obtained.

First quartile = 44 Third quartile = 68 Standard deviation = 9 Range = 55

Mean = 51 Median = 54 Mode = 71 Midrange = 64

Answer each of the following; show all work.

- What score was earned by more students than any other score? Why?

- What was the highest score earned on the exam?

- What was the lowest score earned on the exam?

- According to Chebyshev's Theorem, how many students scored between 24 and 78?

- Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 24 and 78? (

2. Answer the following:

- If the correlation coefficient is -0.79, what is the sign of the slope of the regression line?

- As the correlation coefficient increases from 0.21 to 0.28, do the points of the scatter plot move toward the regression line, or away from it?

3. A math test was given; a sample of the scores yielded the following results:

32, 17, 83, 42, 32, 70, 94, 88, 27, 10, 76, 27, 98, 7, 7, 53

Find the range, standard deviation, and variance for the sample scores.

4. In terms of the mean and standard deviation:

- What does it mean to say that a particular value of x has a standard score of +2.1?

- What does it mean to say that a particular value of x has a z-score of -1.2?

5. A student scored 99 percent on a test, and was in the 97th percentile. Explain these two numbers.

6. An animal trainer obtained the following sample data (Table A) in a study of reaction times of dogs (in seconds) to a specific stimulus. He then selected another group of dogs that were much older than the first group and measure their reaction times to the same stimulus. The sample data is shown in Table B.

Table A Table B

Classes Frequency Classes Frequency

2.3-2.9 24 2.3-2.9 4

3.0-3.6 24 3.0-3.6 21

3.7-4.3 14 3.7-4.3 8

4.4-5.0 5 4.4-5.0 0

5.1-5.7 4 5.1-5.7 24

5.8-6.4 16 5.8-6.4 20

Find the variance and standard deviation for the two distributions above. Compare the variation of the data sets. Decide if one data set is more variable than the other.

7. You are given the following data.

Number of Absences Final

Grade

0 93

1 92

2 79

2 70

3 72

3 68

4 64

5 56

6 40

? Make a scatter plot for the data;

? Find the correlation coefficient for the data.

? Find the equation for the regression line for the data, and predict the final grade of a student who misses 3.5 days.

8. Indicate which would be the independent variable and which would be the dependent variable:

Distance = dependent, Travel time required = independent

Distance = independent, Travel time required = dependent

9. Indicate which would be the independent variable and which would be the dependent variable:

Salary = dependent, Income Tax Rate = independent

Salary = independent, Income Tax Rate = dependent