3 Describing Data Numerically

Section 3.3 Exercises

CLARIFYING THE CONCEPTS

Question 3.214

1. Explain why the formula for the mean of grouped data will provide an estimate only and not the exact value of the mean if the data were not grouped. (p. 149)

3.3.1

These formulas will provide only estimates because we will not know the exact data values.

Question 3.215

2. Describe how the weighted mean is calculated. (p. 149)

Question 3.216

3. Suppose we calculate the weighted mean of the following data: 2, 7, 4. Let each of the weights equal 1. To what measure of center from Section 3.1 does this weighted mean simplify when all the weights equal 1? (p. 149)

3.3.3

The mean

PRACTICING THE TECHNIQUES

CHECK IT OUT!

To do	Check out	Topic
Exercises 4–8	Example 17	Weighted mean
Exercises 9–14	Example 18	Estimated mean for grouped data
Exercises 15–20	Example 19	Estimated variance and standard deviation for grouped data

For Exercises 4–8, the data values and weights are provided. Find the weighted mean.

Question 3.217

4. , ; ; , , .

Question 3.218

5. , , ; , , .

3.3.5

80.5

Question 3.219

6. , , ; , , .

Question 3.220

7. , , , , ; , .

3.3.7

2.45

Question 3.221

8. , , , ; , , , .

For Exercises 9–14, the frequency distribution is provided for a particular variable. Do the following:

Find the class midpoints.
Calculate the product of each class frequency with its midpoint.
Find the sum of the frequencies, , and the sum of the products .
Divide by to find the estimated mean of the variable, .

153

Question 3.222

Class	Frequency
	2
	10
	13
	5

3.3.9

(a)-(c)

Class	Frequency	Midpoint	Product ·
	2	0.5	1.0
	10	1.5	15.0
	13	2.5	32.5
	5	3.5	17.5
Total

(d)

Question 3.223

10.

Class	Frequency
	3
	7
	7
	3

Question 3.224

11.

Class	Frequency
	10
	20
	30
	20
	10

3.3.11

(a)–(c)

Class	Frequency	Midpoint	Product ·
	10	1	10
	20	3	60
	30	5	150
	20	7	140
	10	9	90
Total

(d)

Question 3.225

12.

Class	Frequency
	5
	10
	15
	20
	20

Question 3.226

13.

Class	Frequency
	100
	150
	200
	250
	300
	350
	400
	450

3.3.13

(a)–(c)

Class	Frequency	Midpoint	Product
	100	2.5	250
	150	7.5	1,125
	200	12.5	2,500
	250	17.5	4,375
	300	25	7,500
	350	40	14,000
	400	75	30,000
	450	150	67,500
Total

(d)

Question 3.227

14.

Class	Frequency
	15
	10
	5
	4
	4
	2
	1
	1

For Exercises 15–20, find the estimated variance and standard deviation for the frequency distribution given in the indicated Exercise.

Question 3.228

15. Exercise 9.

3.3.15

Variance: ; Standard deviation:

Question 3.229

16. Exercise 10.

Question 3.230

17. Exercise 11.

3.3.17

Variance: ; Standard deviation:

Question 3.231

18. Exercise 12.

Question 3.232

19. Exercise 13.

3.3.19

Variance: ; Standard deviation:

Question 3.233

20. Exercise 14.

APPLYING THE CONCEPTS

Question 3.234

dupageage

21. Dupage County age groups. The Census Bureau reports the following frequency distribution of population by age group for Dupage County, Illinois, for residents who are less than 65 years old.

Class	Residents
	63,422
	240,629
	540,949

Find the class midpoints.
Find the estimated mean age of residents of Dupage County.
Find the estimated variance and standard deviation of ages.

3.3.21

(a)

Age	Frequency	Midpoints
0–4.99	63,422	2.5
5–17.99	240,629	11.5
18–64.99	540,949	41.5

(b) Estimated mean = 30.0298 years (c) Estimated standard deviation = 15.455909 years; estimated variance = 238.8851 years squared

Question 3.235

browardhouse

22. Broward County House Values. Table 19 gives the frequency distribution of the dollar value of the owner-occupied housing units in Broward County, Florida.

Table 3.77: Table 19 Broward County house values

Class (1000s)	Housing units
	5,430
	90,605
	90,620
	54,295
	34,835
	15,770
	5,595

Find the class midpoints.
Find the estimated mean dollar value for housing units in Broward County.
Find the estimated variance and standard deviation of the dollar value.

Question 3.236

lightningdeath

23. Lightning Deaths. Table 20 gives the frequency distribution of the number of deaths due to lightning nationwide over a 67-year period. Find the estimated mean and standard deviation of the number of lightning deaths per year.

Table 3.78: Table 20 Lightning deaths

Class	Years
	13
	21
	10
	6
	10
	7

Table 3.78: Source: National Oceanic and Atmospheric Administration.

3.3.23

Estimated mean = 135.5224; estimated standard deviation = 95.6874

154

Question 3.237

24. Calculating a Course Grade. An introductory statistics syllabus has the following grading system. The weekly quizzes are worth a total of 25% toward the final course grade. The midterm exam is worth 32%; the final exam is worth 33%; and attendance/participation is worth 10% toward the final course grade. Anthony's weekly quiz average is 70. He got an 80 on the midterm and a 90 on the final exam. He got a 100 for attendance/participation. Calculate Anthony's final course grade.

Question 3.238

compwage

25. Wages for Computer Managers. The U.S. Bureau of Labor Statistics (BLS) publishes wage information for various occupations. For the occupation “computer and information systems management,” Table 21 gives the wages reported by the BLS for the top-paying states. Find the weighted mean wage across all five states, using the employment figures as weights.

Table 3.79: Table 21 Wages for computer managers

State	Employment	Hourly mean wage
New Jersey	12,380	$60.32
New York	18,580	$60.25
Virginia	9,540	$59.39
California	35,550	$57.98
Massachusetts	10,130	$55.95

3.3.25

$58.72

Question 3.239

26. Salaries of Scientists and Engineers. The National Science Foundation compiles statistics on the annual salaries of full-time employed doctoral scientists and engineers in universities and four-year colleges. The mean annual salary for the fields of science, engineering, and health are $67,000, $82,200, and $70,000, respectively. Suppose we have a sample of 10 professors, 5 of whom are in science, 2 in engineering, and 3 in health, and each of whom is making the mean salary for his or her field. Find the weighted mean salary of these 10 professors.

Question 3.240

27. Challenge Exercise. Assign the weights, , to show that the formula for the sample mean from Section 3.1, , is a special case of the formula for the weighted mean, .

3.3.27

If w_i = 1 for all , then the weighted mean formula will be equivalent to the formula for the sample mean.

BRINGING IT ALL TOGETHER

Wait Times at Los Angeles Airport. Use the following table for Exercises 28–33. The data represent the number of passengers whose flights were delayed at the Tom Bradley Terminal of Los Angeles Airport (LAX), on July 2, 2014, between 4 P.M. and 5 P.M. Counts are given based on how long their flights were delayed.

Delay (minutes)	Passengers
	665
	551
	497
	399
	355
	27

Table 3.80: Source: U.S. Customs and Border Protection: awt.cbp.gov.

Question 3.241

28. Find the delay midpoints.

Question 3.242

29. Construct a table similar to Table 17, showing the frequencies, , the midpoints, , the products, , the sum of the frequencies, , and the sum of the products, .

3.3.29

Delay (minutes)	Frequency	Midpoint	Product
0 to < 16	665	8	5,320
16 to < 31	551	23.5	12,948.5
31 to < 46	497	38.5	19,134.5
46 to < 61	399	53.5	21,346.5
61 to < 91	355	76	26,980
91 to < 120	27	105.5	2,848.5
Total

Question 3.243

30. Use the quantities from Exercise 29 to calculate the estimated mean delay time.

Question 3.244

31. Extend your table from Exercise 29 so that it is similar to Table 18, including columns for , , and . Calculate .

3.3.31

Delay (minutes)	Midpoint	Frequency
0 to < 16	8	665	35.52	–27.52	503,638.016
16 to < 31	23.5	551	35.52	–12.02	79,608.7004
31 to < 46	38.5	497	35.52	2.98	4,413.5588
46 to < 61	53.5	399	35.52	17.98	128,988.8796
61 to < 91	76	355	35.52	40.48	581,713.792
91 to < 120	105.5	27	35.52	69.98	132,224.4108
Total

Question 3.245

32. Use the statistics from Exercise 31 to compute the estimated variance.

Question 3.246

33. Calculate the estimated standard deviation of delay times.

3.3.33

23.9502

WORKING WITH LARGE DATA SETS

Financial Experts versus the Darts. This set of exercises examines how close the estimated mean, variance, and standard deviation are to their true values. Use the Darts data set from the Chapter 3 Case Study for Exercises 34–37.

Question 3.247

darts

34. Use the following classes to construct a frequency distribution for the professionals, Darts, and the DJIA data sets.

Class

Question 3.248

darts

35. Use the frequency distribution from Exercise 34 to calculate the estimated mean stock price change for the professionals, Darts, and the DJIA data sets.

3.3.35

Professionals: 13.25

Darts: 5.5

DJIA: 7.75

Question 3.249

darts

36. Use the information from the two previous exercises to compute the estimated variance and standard deviation for the stock price changes for the professionals, Darts, and the DJIA data sets.

Question 3.250

darts

37. Using technology, find the mean, variance, and standard deviation for the professionals, Darts, and the DJIA data sets. Calculate the difference between the estimated values and the actual values.

3.3.37

Professionals:

Mean: 10.95

Variance:

Standard deviation:

Darts:

Mean:

Variance:

Standard deviation:

DJIA:

Mean:

Variance:

Standard deviation:

WORKING WITH LARGE DATA SETS

Year-by-year age distribution. Open the Age Distribution 100 data set, and use it for Exercises 38–42. This data set shows the year-by-year age distribution of Americans under age 100, as reported by the U.S. Census Bureau, for 2011. Use technology to answer the following:

Question 3.251

agedistribution 100

38. How many tiny tots have yet to reach their first birthday?

Question 3.252

agedistribution 100

39. Find the mean age of Americans under 100.

3.3.39

37.30 years

Question 3.253

agedistribution 100

40. Calculate the estimated standard deviation of Americans under 100.

Question 3.254

agedistribution 100

41. Use the Empirical Rule (see Section 3.2) to find two age values between which lie about 68% of the ages of all Americans under 100.

3.3.41

14.6 years and 60 years

Question 3.255

agedistribution 100

42. Compute the actual proportion between the age values found in the previous exercise. Compare the actual number to the estimate in the previous exercise.