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# Distributions and Central Tendency

1. True or false? The median is the only measure of central tendency that can be used to describe data at the nominal level of measurement.

2. True or false? If a vertical line is drawn through the middle of a distribution and both halves are approximately mirror images, the distribution may still be classified as symmetric.

3. Determine whether the shape of the distribution represented by the histogram is symmetric, uniform, skewed left, skewed right, or none of these.

4. Match the distribution with one of the graphs below. Select the letter for the appropriate graph. The frequency distribution is the weights of students in two classes at a local high school.
a. b.
c. d.

For problems 5-7, find the mean, median, and mode of the data, if possible. If any of these measures cannot be found, explain why.

5. Cholesterol
The cholesterol levels of a sample of 10 female employees.
162 210 180 175 170 199 215 167 275 178

6. Air Craft
The number of aircraft that 15 airlines have in their fleets.
84 135 44 587 298 722 60 27 155 14 487 44 359 422 26

The responses of a sample of 1366 young adults who were asked what surprised them the most as they began to live on their own.
Amount of first salary: 89
Number of decisions: 183
Paying bills: 326
Trying to find a job: 138
Money needed: 300
Trying to save: 240
Difficulty in breaking away from parents: 90

8. Find the weighted mean of the data
For the month of July, a checking account has a balance of \$275.29 for 14 days, \$758.48 for 6 days, \$723.84 for 8 days, and \$279.40 for 3 days. What is the account's mean daily balance for July?

Section 2.4
Use the data set representing a sample below for problems 9-12. If you have not done so already, review the 1-var stats function on the TI83/84 calculator.

38 35 30 32 16
28 15 56 42 24
39 44 46 22 39
29 55 25 64 25

9. What is the range?
10. What is the mean?
11. What is the variance?
12. What is the standard deviation?

Use the data sets below to answer questions 13-15.

Data set A: 100 120 101 119 102 118 118 102 119 101
Data set B: 100 120 114 106 113 107 113 107 112 108

13. Select the letter of the appropriate data set. Without calculating the standard deviation, which data set has the greatest standard deviation?
14. Select the letter of the appropriate data set. Without calculating the standard deviation, which data set has the least standard deviation?
15. Explain the reason for your decisions in questions 13 and 14.

16. Tree Heights
You study the heights of trees on two wood lots. Lot A has trees with = 22 feet and = 2 feet and Lot B has trees with heights, such that = 22 feet and = 6 feet.
Which lot is most likely to have trees greater than 28 feet high?

17. Use the empirical rule to solve the problem.
The mean annual income at a small company is \$39,400, with a standard deviation of \$3,325.00. Between what two values do about 95% of the salaries lie?

Use the following information to answer question 18.
Coefficient of Variation The coefficient of variation, CV, describes the standard deviation as a percent of the mean. Because it has no units, you can use the coefficient of variation to compare the variability of data with different units. (Note: Either the sample standard deviation, s, or the population standard deviation, may be used for calculating the CV. Use the appropriate value according to the problem statement indicating whether it refers to a population or a sample.)

CV =
standard deviation
Mean  100%

18. Find the coefficient of variation for the data set given for problems 9-12.

Section 2.5

19. True or False? If a data set has Q1 = 60, Q2 = 85 and Q3 = 98, then a data value of 145 would be considered an outlier.

20. True or False? 50% of the data in a data set with quantitative data is between Q1 and Q3.

Use the data set below to answer questions 30-33.

187 238 235 218 248 230 245 209 216
233 228 214 236 250 222 266 231 219
207 219 246 229 226 234 239

30. What is Q1?
31. What is Q2?
32. What is Q3?
33. Draw a box-and-whisker plot that represents the data set.

Data Analysis: Use the following information to answer questions 1-3. A consumer testing
service obtained the following miles per gallon in five test runs performed with three types of
compact cars.

Run 1 Run 2 Run 3 Run 4 Run 5
Car A: 28 32 28 30 34
Car B: 31 29 31 29 31
Car C: 29 32 28 32 30

1. The manufacturer of Car A wants to advertise that its car performed best in this test. Which measure of central tendency - mean, median, or mode - should be used for its claim? Explain your reasoning.
2. The manufacturer of Car B wants to advertise that its car performed best in this test. Which measure of central tendency - mean, median, or mode - should be used for its claim? Explain your reasoning.
3. The manufacturer of Car C wants to advertise that its car performed best in this test. Which measure of central tendency - mean, median, or mode - should be used for its claim? Explain your reasoning.

Data Analysis: Use the following information to answer questions 4-7.
Students in an experimental psychology class did research on depression as a sign of stress. A test was administered to a sample of 30 students. The scores are given.

44 51 11 90 76 36 64 37 43 72 53 62 36 74 51
72 37 28 38 61 47 63 36 41 22 37 51 46 85 13

4. Find the mean of the data.
5. Find the median of the data.
6. Draw a stem-and-leaf plot for the data using one row per stem. Mark the median and mean on the display.
7. Describe the shape of the distribution.

Use the following information to answer questions 8-11.
The key for all of the stem-and leaf plots below is: Key: 4j1 = 41.

(a) 0 9 (b) 0 9 (c) 0
1 58 1 5 1 5
2 3377 2 333777 2 33337777
3 25 3 5 3 5
4 1 4 1 4

8. Which data set has the greatest standard deviation?
9. Which data set has the least standard deviation?
10. How are the data sets the same?
11. How are the data sets different?

12. Chebychev's Theorem
The mean time in a women's 400-meter dash is 57.07 seconds, with a standard deviation of 1.05. Apply Chebychev's Theorem to the data using k = 2. Interpret the results.

Use the following information to answer questions 13-19.
Scaling Data Sample
Annual salaries (in thousands of dollars) for employees at a company are listed.

42 36 48 51 39 39 42 36 48 33 39 42 45

13. What is the sample mean?
14. What is the sample standard deviation?

Each employee in the sample is given a 5% raise.
15. What is the sample mean for the revised data set?
16. What is the sample standard deviation for the revised data set?

Calculate the monthly salary be dividing each salary in the original sample by 12.
17. What is the sample mean for the revised data set?
18. What is the sample standard deviation for the revised data set?
19. What do you conclude from the results in problems 13-18 above?

Use the following information to answer questions 20-26.
Shifting Data Sample
Annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40 37 49 34 38 43 47

20. What is the sample mean?
21. What is the sample standard deviation?

Each employee in the sample is given a \$1000 raise.
22. What is the sample mean for the revised data set?
23. What is the sample standard deviation for the revised data set?

Each employee in the sample takes a \$2000 salary cut from their salary reported in the
original sample.
24. What is the sample mean for the revised data set?
25. What is the sample standard deviation for the revised data set?
26. What do you conclude from the results in problems 20-25 above?

Use the following information to answer questions 27-34.
Hourly Earnings
The hourly earnings (in dollars) of a sample of 25 railroad equipment manufacturers.

15.60 18.75 14.60 15.80 14.35 13.90 17.50 17.55 13.80
14.20 19.05 15.35 15.20 19.45 15.95 16.50 16.30 15.25
15.05 19.10 15.20 16.22 17.75 18.40 15.25

27. What is Q1?
28. What is Q2?
29. What is Q3?
30. Draw a box-and-whisker plot to display the data.
31. Are there outliers in the data set? Explain your reasons (See the next to the last paragraph
page 104).
32. About 75% of the manufacturers made less than what amount per hour?
33. What percent of the manufacturers made more than \$15:80 per hour?
34. If you randomly selected one manufacturer from the sample, what is the likelihood that the
manufacturer made less than \$15:80 per hour?

#### Solution Preview

Hi there,

Thanks for letting me work on your post. I've included my explanation in the word document.

Thanks.
Tony

----------------

1. True or false? The median is the only measure of central tendency that can be used to describe data at the nominal level of measurement.
False (the mode is the best measure in this case).

2. True or false? If a vertical line is drawn through the middle of a distribution and both halves
are approximately mirror images, the distribution may still be classified as symmetric.
TRUE.

3. Determine whether the shape of the distribution represented by the histogram is symmetric,
uniform, skewed left, skewed right, or none of these.
Please provide the histogram so that a correct choice could be selected.

4. Match the distribution with one of the graphs below. Select the letter for the appropriate
graph. The frequency distribution of the weights of students in two classes at a local high
school.
a. b. c. d.

For problems 5-7, find the mean, median, and mode of the data, if possible. If any of
these measures cannot be found, explain why.

5. Cholesterol
The cholesterol levels of a sample of 10 female employees.
162 210 180 175 170 199 215 167 275 178

The ordered data are:
162
167
170
175
178
180
199
210
215
275

Mean=sum/10=193.1
Median=(178+180)/2=179.
There is no mode since each number only appears once.

6. Air Craft
The number of aircraft that 15 airlines have in their fleets.
84 135 44 587 298 722 60 27 155 14 487 44 359 422 26
The ordered data are:
14
26
27
44
44
60
84
135
155
298
359
422
487
587
722

The mode is 44.
Median=8th data=135.
Since the variable is a discrete variable, the mean does not carry any particular meaning.

The responses of a sample of 1366 young adults who were asked what surprised them the most as they began to live on their own.

Amount of first salary: 89
Number of decisions: 183
Paying bills: 326
Trying to find a job: 138
Money needed: 300
Trying to save: 240
Difficulty in breaking away from parents: 90

This is nominal variable. The best measure of central tendency is mode. In this case, the mode is paying bills since it appears 326 times which is the highest among all the frequencies.

8. Find the weighted mean of the data
For the month of July, a checking account has a balance of \$275.29 for 14 days, \$758.48 for 6 days, \$723.84 for 8 days, and \$279.40 for 3 days. What is the account's mean daily balance for July?

Mean daily balance=275.29*14/31+758.48*6/31+723.84*8/31+279.40*3/31=484.96 dollars.

Section 2.4
Use the data set representing a sample below for problems 9-12. If you have not done so
already, review the 1-var stats function on the TI83/84 calculator.

38 35 30 32 16
28 15 56 42 24
39 44 46 22 39
29 55 25 64 25

9. What is the range?
Range=max-min=64-15=49.
10. What is the mean?
Mean=Sum/20=35.2
11. What is the variance?
Variance=((15-35.2)^2+(16-35.2)^2+(22-35.2)^2+...+(64-35.2)^2)/(20-1)=175.9579.
12. What is the standard deviation?
Standard deviation=sqrt(variance)=sqrt(175.9579)=13.26491.

Use the data sets below to answer questions 13-15.

Data set A: 100 120 101 119 102 118 118 102 119 101
Data set B: 100 120 114 106 113 107 113 107 112 108

13. Select the letter of the ...

#### Solution Summary

The solution provides a detailed explanation of how to solve multiple statistics problems including box plot, stem-and-leaf plot and others.

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