07 Jan The data are skewed

MATH399 Applied Managerial Statistics

Week 1 Assignment Skewness and Standard Deviation

QuestionBased on the z-scores calculated above for Stephan’s electric bills in IL and FL, in which city is his electric bill higher, when compared to their respective distributions?

Illinois

Florida

Stephan’s electric bills in both states are comparable.

There is not enough information for a comparison.

QuestionWhen Stephan moved from Illinois to Florida, his average monthly electric bill increased from $83 to $102. He is curious to know whether his IL or FL electric bill is relatively more or less expensive, when compared to the distribution of electric bills for each state. In Illinois, the mean monthly electric bill is $85, with a standard deviation of $3.20. In Florida, the mean monthly electric bill is $105, with a standard deviation of $4.00.

Compute the z-scores for Stephan’s IL and FL electric bills. Round to three decimal places if necessary.

QuestionBased on the z scores found above, on which college entrance exam did Seam perform better, compared to the national distributions for each test?

QuestionNational results for the SAT test show that for college-bound seniors the average combined SAT Writing, Math and Verbal score is 1500 with a standard deviation of 250. National results for the ACT test show that for college-bound seniors the average composite ACT score is 20.4 with a standard deviation of 4.8.

Sean took both the SAT and the ACT college entrance exams. His SAT score was 1700 and his ACT score was 24. He wants to know on which test he performed better.

Find the z-scores for his result on each exam.

QuestionA math class’s mean test score is 88.4. The standard deviation is 4.0. If Kimmie scored 85.9, what is her z-score?

?1.6

?0.625

0.625

1.6

Using Z-Scores to Compare Data

Key Terms

• Z-score: a measure of how far a point is from the mean, measured in standard deviations, denoted by the letter z

Z-score is also referred to as the Test statistic

QuestionBased on the z scores found above, is Kathy or Linda’s starting salary higher, when compared to the salary distributions of each company?_

QuestionKathy and Linda both accepted new jobs at different companies. Kathy’s starting salary is $31,500and Linda’s starting salary is $33,000. They are curious to know who has the better starting salary, when compared to the salary distributions of their new employers.

A website that collects salary information from a sample of employees for a number of major employers reports that Kathy’s company offers a mean salary of $42,000 with a standard deviation of $7,000. Linda’s company offers a mean salary of $45,000 with a standard deviation of $6,000.

Find the z-scores corresponding to each woman’s starting salary.

QuestionThe following data values represent the daily amount spent by a family each day during a 7 day summer vacation.

Find the standard deviation of this data set:

$96,$125,$80,$110,$75,$100,$121

Round the final answer to one decimal place.

QuestionThe following data values represent the daily amount spent by a family each day during a 7 day summer vacation.

Find the variance of this dataset:

$96,$125,$80,$110,$75,$100,$121

• Round the final answer to one decimal place.

QuestionThe following data values represent the daily amount spent by a family each day during a 7 day summer vacation.

$96,$125,$80,$110,$75,$100,$121

To determine the “spread” of the data, would you employ calculations for the sample standard deviation, or population standard deviation for this data set?

Use calculations for sample standard deviation

Use calculations for population standard deviation

QuestionA food processing plant fills snack-sized bags of crackers. The mean number of crackers in each bag is 22 and the standard deviation is 2. The factory supervisor selects one bag that contains 24 crackers.

Which of the following statements is true?

The number of crackers in the supervisor’s bag is 2 standard deviations to the right of the mean.

The number of crackers in the supervisor’s bag is 2 standard deviations to the left of the mean.

The number of crackers in the supervisor’s bag is 1 standard deviation to the right of the mean.

The number of crackers in the supervisor’s bag is 1 standard deviation to the left of the mean.

QuestionWhich of the following frequency tables show a skewed data set? Select all answers that apply.

•

Value Frequency

0 2

1 11

2 30

3 22

4 15

5 12

6 6

7 1

8 1

• ________________________________________

•

Value Frequency

4 1

5 2

6 3

7 7

8 19

9 17

10 17

11 15

12 12

13 4

14 1

15 2

• ________________________________________

•

Value Frequency

13 1

14 6

15 9

16 15

17 27

18 28

19 10

20 4

• ________________________________________

•

Value Frequency

3 1

4 0

5 1

6 5

7 9

8 12

9 12

10 18

11 12

12 17

13 11

14 0

15 1

16 0

17 1

QuestionWhich of the data sets represented by the following histograms has the largest standard deviation?

A histogram has a horizontal axis from 0.00 to 50.00 in increments of 2.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 2.00 starting at the horizontal axis value 22.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 14.00, 80; 16.00, 95; 18.00, 45.

A histogram has a horizontal axis from 0.00 to 50.00 in increments of 2.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 2.00 starting at the horizontal axis value 12.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 12.00, 67; 14.00, 80; 16.00, 95; 18.00, 75; 20.00, 75.

A histogram has a horizontal axis from 0.00 to 50.00 in increments of 2.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 2.00 starting at the horizontal axis value 18.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 18.00, 50; 20.00, 70; 22.00, 74; 24.00, 78; 26.00, 83; 28.00, 81; 30.00, 80; 32.00, 74; 34.00, 60.

A histogram has a horizontal axis from 0.00 to 50.00 in increments of 2.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 2.00 starting at the horizontal axis value 8.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 8.00, 49; 10.00, 51; 12.00, 51; 14.00, 53; 16.00, 51; 18.00, 51; 20.00, 48; 22.00, 49; 24.00, 51; 26.00, 53; 28.00, 50; 30.00, 53; 32.00, 51; 34.00, 53; 36.00, 52; 38.00, 46.

QuestionIf the mean of a data set is 32 and the median is 36, which of the following is most likely?

The data are skewed to the left.

The data are skewed to the right.

The data are symmetric.

QuestionWhich of the following box-and-whisker plots shows a skewed data set? Select all answers that apply.

A horizontal box-and-whisker plot is above a horizontal axis labeled from 0 to 18 in increments of 2. The box-and-whisker plot has the following five-number summary: 2, 8, 9, 10, and 16. All values are approximate. The part of the box at point 9 is represented with a dotted line.

A horizontal box-and-whisker plot is above a horizontal axis labeled from 0 to 20 in increments of 5. The box-and-whisker plot has the following five-number summary: 3, 9, 10, 11, and 18. All values are approximate. The part of the box at point 10 is represented with a dotted line.

A horizontal box-and-whisker plot is above a horizontal axis labeled from 0 to 16 in increments of 2. The box-and-whisker plot has the following five-number summary: 1, 3, 4, 6, and 15. All values are approximate. The part of the box at point 4 is represented with a dotted line.

A horizontal box-and-whisker plot is above a horizontal axis labeled from 0 to 20 in increments of 5. The box-and-whisker plot has the following five-number summary: 4, 14, 16, 19, and 20. All values are approximate. The part of the box at point 16 is represented with a dotted line.

Question Which of the data sets represented by the following histograms has the largest standard deviation?

A histogram has a horizontal axis from 0.00 to 90.00 in increments of 10.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 10.00 starting at the horizontal axis value 30.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 30.00, 51; 40.00, 60; 50.00, 87; 60.00, 71.

A histogram has a horizontal axis from 0.00 to 90.00 in increments of 10.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 10.00 starting at the horizontal axis value 30.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 30.00, 31; 40.00, 42; 50.00, 48; 60.00, 34; 70.00, 25.

A histogram has a horizontal axis from 0.00 to 90.00 in increments of 10.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 10.00 starting at the horizontal axis value 40.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 40.00, 82; 50.00, 76; 60.00, 70; 70.00, 98; 80.00, 76.

A histogram has a horizontal axis from 0.00 to 90.00 in increments of 10.00 and a vertical axis from 0 to 100 in increments of 25. The histogram has vertical bars of width 10.00 starting at the horizontal axis value 0.00. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 0.00, 55; 10.00, 32; 20.00, 34; 30.00, 50; 40.00, 61; 50.00, 87; 60.00, 70; 70.00, 62; 80.00, 46; 90.00, 48.

QuestionBased on the z-scores calculated above for Martin and Lawrence, whose salary is higher, when compared to their companies?

Martin

Lawrence

Neither salary is higher, when compared to the company salaries.

There is not enough information for a comparison.

QuestionMartin and Lawrence want to know whose salary is higher, when compared to the distribution of salaries at their jobs. The table shows their salaries, as well as the mean salary and standard deviation for each of the companies at which they work.

Name Annual Salary Company

Mean Salary Company

Standard Deviation

Martin $44,000 $40,000 $6,400

Lawrence $46,000 $41,500 $7,200

Find the z-scores corresponding to each man’s salary. Round to three decimal places, if necessary.

QuestionUsing the following set of data (the same as in the previous problem), find the population standard deviation: 8, 9, 5, 12, 6.

The population variance of this data set is 6.

QuestionFind the population variance of the following set of data:

8, 9, 5, 12, 6.

QuestionA group of five friends are long-distance runners who will be running in an upcoming marathon. To prepare for the marathon, each of the friends recorded the number of miles they ran each week for 12weeks. The results of the friends’ training are shown in the data set provided below. Which of the friends was the most consistent with the number of miles run each week during the training? Hint: You should not need to compute the standard deviation for each friend.

Ashton

Caleb

Jordan

Kyle

Noah

24

35

22

15

13

34

36

39

26

29

25

34

26

24

18

31

35

41

25

31

19

37

38

27

15

33

35

37

24

19

27

34

40

25

27

29

36

27

26

32

36

35

21

32

15

HelpCopy to ClipboardDownload CSV

Ashton

Caleb

Jordan

Kyle

Noah

QuestionYou are told that a data set has a median of 64 and a mean of 52. Which of the following is a logical conclusion?

The data are skewed to the left.

The data are skewed to the right.

The data are symmetric.