STAT 200 Week 7 Homework Problems

10.1.2

Table #10.1.6 contains the value of the house and the amount of rental income in a year that the house brings in (“Capital and rental,” 2013). Create a scatter plot and find a regression equation between house value and rental income. Then use the regression equation to find the rental income a house worth $230,000 and for a house worth $400,000. Which rental income that you calculated do you think is closer to the true rental income? Why?

Table #10.1.6: Data of House Value versus Rental

Value

Rental

Value

Rental

Value

Rental

Value

Rental

81000

6656

77000

4576

75000

7280

67500

6864

95000

7904

94000

8736

90000

6240

85000

7072

121000

12064

115000

7904

110000

7072

104000

7904

135000

8320

130000

9776

126000

6240

125000

7904

145000

8320

140000

9568

140000

9152

135000

7488

165000

13312

165000

8528

155000

7488

148000

8320

178000

11856

174000

10400

170000

9568

170000

12688

200000

12272

200000

10608

194000

11232

190000

8320

214000

8528

208000

10400

200000

10400

200000

8320

240000

10192

240000

12064

240000

11648

225000

12480

289000

11648

270000

12896

262000

10192

244500

11232

325000

12480

310000

12480

303000

12272

300000

12480

Value

Rental

(x-sample mean of x)^2

(y-sample mean of y)^2

(x-sample mean of x)(y-sample mean of y)

174375

9611.333

2.26936E+11

230247402.7

5527756000

Mean x

Mean y

SSx

SSy

SSxy

Slope = b = SSxy/SSx = 5527756000/2.26936E+11 = 0.024

y-intercept = a = -bx = 9611.333-0.024(174375) = 5426.333

Regression equation: y-intercept = 5426.333+0.024x

House value at $230,000 = 5426.333+0.024(230000) = $10,946.333

House value at $400,000 = 5426.333+0.024(400000) = $15,026.333

Rental income of $10,946.333 for a house valued at $230K is closer to the true rental income because the values fall within the range of the original values.

10.1.4

The World Bank collected data on the percentage of GDP that a country spends on health expenditures (“Health expenditure,” 2013) and also the percentage of women receiving prenatal care (“Pregnant woman receiving,” 2013). The data for the countries where this information are available for the year 2011 is in table #10.1.8. Create a scatter plot of the data and find a regression equation between percentage spent on health expenditure and the percentage of women receiving prenatal care. Then use the regression equation to find the percent of women receiving prenatal care for a country that spends 5.0% of GDP on health expenditure and for a country that spends 12.0% of GDP. Which prenatal care percentage that you calculated do you think is closer to the true percentage? Why?

Table #10.1.8: Data of Health Expenditure versus Prenatal Care

Health Expenditure (% of GDP)

Prenatal Care (%)

9.6

47.9

3.7

54.6

5.2

93.7

5.2

84.7

10.0

100.0

4.7

42.5

4.8

96.4

6.0

77.1

5.4

58.3

4.8

95.4

4.1

78.0

6.0

93.3

9.5

93.3

6.8

93.7

6.1

89.8

Health Expenditure (% of GDP)

Prenatal Care (%)

(x-mean)^2

(y-mean)^2

(x-mean x)(y-mean y)

6.126667

79.91333

56.72933

5318.417

94.20466667

mean

mean

SSx

Ssy

Ssxy

Slope = b = SSxy/SSx = 94.205/56.729 = 1.66

y-intercept = a = y ?-bx = 79.913-1.661(6.127) = 69.74%

Regression equation: y-intercept = 69.74+1.66x

prenatal care for a country that spends 5.0% of GDP = 69.74+1.66(5) = 78.04%

prenatal care for a country that spends 12.0% of GDP = 69.74+1.66(12) = 89.66%

The prenatal care for a country that spends 5% of GDP is closer to the true percentage because it is closer to the regression line

10.2.2

Table #10.1.6 contains the value of the house and the amount of rental income in a year that the house brings in (“Capital and rental,” 2013). Find the correlation coefficient and coefficient of determination and then interpret both.

Table #10.1.6: Data of House Value versus Rental

Value

Rental

Value

Rental

Value

Rental

Value

Rental

81000

6656

77000

4576

75000

7280

67500

6864

95000

7904

94000

8736

90000

6240

85000

7072

121000

12064

115000

7904

110000

7072

104000

7904

135000

8320

130000

9776

126000

6240

125000

7904

145000

8320

140000

9568

140000

9152

135000

7488

165000

13312

165000

8528

155000

7488

148000

8320

178000

11856

174000

10400

170000

9568

170000

12688

200000

12272

200000

10608

194000

11232

190000

8320

214000

8528

208000

10400

200000

10400

200000

8320

240000

10192

240000

12064

240000

11648

225000

12480

289000

11648

270000

12896

262000

10192

244500

11232

325000

12480

310000

12480

303000

12272

300000

12480

Correlation coefficient: r = SSxy/?SSxSSy = 5527756000/?2.26936E+11*230247402.7 = 0.7647

0.7647 is close to 1, therefore there is a strong, positive correlation

Coefficient of determination: r^2 = (r)^2 = (0.7647)^2 = 0.5848

Thus, 58.48% of the variation in rental is explained to the linear relationship between house value versus rental. The other 41.52% of the variation is due to other factors.

10.2.4

The World Bank collected data on the percentage of GDP that a country spends on health expenditures (“Health expenditure,” 2013) and also the percentage of women receiving prenatal care (“Pregnant woman receiving,” 2013). The data for the countries where this information is available for the year 2011 are in table #10.1.8. Find the correlation coefficient and coefficient of determination and then interpret both.

Table #10.1.8: Data of Health Expenditure versus Prenatal Care

Health Expenditure (% of GDP)

Prenatal Care (%)

9.6

47.9

3.7

54.6

5.2

93.7

5.2

84.7

10.0

100.0

4.7

42.5

4.8

96.4

6.0

77.1

5.4

58.3

4.8

95.4

4.1

78.0

6.0

93.3

9.5

93.3

6.8

93.7

6.1

89.8

Correlation coefficient: r = SSxy/?SSxSSy = 94.20466667/?56.72933*5318.417 = 0.1715

0.1715 is closer to 0, therefore there is a weak correlation

Coefficient of determination: r^2 = (r)^2 = (0.1715)^2 = 0.0294

Thus, 2.94% of the variation in prenatal care is explained to the weak linear relationship between house value versus rental. The other 97.06% of the variation is due to other factors.

10.3.2

Table #10.1.6 contains the value of the house and the amount of rental income in a year that the house brings in (“Capital and rental,” 2013).

Test at the 5% level for a positive correlation between house value and rental amount.

Table #10.1.6: Data of House Value versus Rental

Value

Rental

Value

Rental

Value

Rental

Value

Rental

81000

6656

77000

4576

75000

7280

67500

6864

95000

7904

94000

8736

90000

6240

85000

7072

121000

12064

115000

7904

110000

7072

104000

7904

135000

8320

130000

9776

126000

6240

125000

7904

145000

8320

140000

9568

140000

9152

135000

7488

165000

13312

165000

8528

155000

7488

148000

8320

178000

11856

174000

10400

170000

9568

170000

12688

200000

12272

200000

10608

194000

11232

190000

8320

214000

8528

208000

10400

200000

10400

200000

8320

240000

10192

240000

12064

240000

11648

225000

12480

289000

11648

270000

12896

262000

10192

244500

11232

325000

12480

310000

12480

303000

12272

300000

12480

Ho:?=0(There is no correlation)

HA:??0(There is a correlation)

or HA:?<0(There is a negative correlation) or HA:?>0(There is a positive correlation)

? level = 0.05

10.3.4

The World Bank collected data on the percentage of GDP that a country spends on health expenditures (“Health expenditure,” 2013) and also the percentage of women receiving prenatal care (“Pregnant woman receiving,” 2013). The data for the countries where this information is available for the year 2011 are in table #10.1.8.

Test at the 5% level for a correlation between percentage spent on health expenditure and the percentage of women receiving prenatal care.

Table #10.1.8: Data of Health Expenditure versus Prenatal Care

Health Expenditure (% of GDP)

Prenatal Care (%)

9.6

47.9

3.7

54.6

5.2

93.7

5.2

84.7

10.0

100.0

4.7

42.5

4.8

96.4

6.0

77.1

5.4

58.3

4.8

95.4

4.1

78.0

6.0

93.3

9.5

93.3

6.8

93.7

6.1

89.8

11.1.2

Researchers watched groups of dolphins off the coast of Ireland in 1998 to determine what activities the dolphins partake in at certain times of the day (“Activities of dolphin,” 2013). The numbers in table #11.1.6 represent the number of groups of dolphins that were partaking in an activity at certain times of days. Is there enough evidence to show that the activity and the time period are independent for dolphins? Test at the 1% level.

Table #11.1.6: Dolphin Activity

Activity

Period

Row

Total

Morning

Noon

Afternoon

Evening

Travel

6

6

14

13

39

Feed

28

4

0

56

88

Social

38

5

9

10

62

Column Total

72

15

23

79

189

11.1.4

A person’s educational attainment and age group was collected by the U.S. Census Bureau in 1984 to see if age group and educational attainment are related. The counts in thousands are in table #11.1.8 (“Education by age,” 2013). Do the data show that educational attainment and age are independent? Test at the 5% level.

Table #11.1.8: Educational Attainment and Age Group

Education

Age Group

Row Total

25-34

35-44

45-54

55-64

>64

Did not complete HS

5416

5030

5777

7606

13746

37575

Competed HS

16431

1855

9435

8795

7558

44074

College 1-3 years

8555

5576

3124

2524

2503

22282

College 4 or more years

9771

7596

3904

3109

2483

26863

Column Total

40173

20057

22240

22034

26290

130794

11.2.4

In Africa in 2011, the number of deaths of a female from cardiovascular disease for different age groups are in table #11.2.6 (“Global health observatory,” 2013). In addition, the proportion of deaths of females from all causes for the same age groups are also in table #11.2.6. Do the data show that the death from cardiovascular disease are in the same proportion as all deaths for the different age groups? Test at the 5% level.

Table #11.2.6: Deaths of Females for Different Age Groups

Age

5-14

15-29

30-49

50-69

Total

Cardiovascular Frequency

8

16

56

433

513

All Cause Proportion

0.10

0.12

0.26

0.52

11.2.6

A project conducted by the Australian Federal Office of Road Safety asked people many questions about their cars. One question was the reason that a person chooses a given car, and that data is in table #11.2.8 (“Car preferences,” 2013).

Table #11.2.8: Reason for Choosing a Car

Safety

Reliability

Cost

Performance

Comfort

Looks

84

62

46

34

47

27

Do the data show that the frequencies observed substantiate the claim that the reasons for choosing a car are equally likely? Test at the 5% level.

Ho: the reason for choosing a car is equally likely

Ha: the reason for choosing a car is not equally likely

Observed frequency = 84+62+46+34+47+27 = 300

Expected value = sum of all frequency/#rows = 300/6 = 50

Percentage of GDP a country spends

9.6 3.7 5.2 5.2 10 4.7 4.8 6 5.4 4.8 4.0999999999999996 6 9.5 6.8 6.1 47.9 54.6 93.7 84.7 100 42.5 96.4 77.099999999999994 58.3 95.4 78 93.3 93.3 93.7 89.8

Health Expenditure %

Pre-natal Care %

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