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

 

Scatterplot with regression equation:

 

Hence line of best fit can be given by:

Y = 0.0244x + 5363.9

Now if we take x = $230,000

Predicted Rental = 0.0244*230000 + 5363.9 = $10,975.90

Again if we take x = $400,000

Predicted Rental = 0.0244*400000 + 5363.9 = $15123.9

It looks like rental income calculated for $230,000 will be much closer to the actual rental. The reason may be that, $230,000 is within the range of original data but $400,000 is outside the scope of original data.

 

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

 

Scatterplot along with line of best fit:

 

The line of best fit is:

Y = 1.6606x + 69.739

Now if we take health expenditure x = 5%,

Predicted Prenatal care = 1.6606 * 5 + 69.739 = 78.042%

Again, if health expenditure x = 12%,

Predicted Prenatal care = 1.6606 * 12 + 69.739 = 89.67%

Itlooks like the prenatal care for 5% health expenditure is much closer to the actual value as 5% is within the range of original data whereas 12% is outside the scope of original data.

 

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

 

I used excel data analysis tool to calculate the correlation coefficient between house value and rental.

 

According to the above output, Pearson correlation coefficient r = 0.764716

It looks like a strong positive linear relation exist between house value and rental.

Coefficient of determination r² = 0.7647² = 0.585

So, 58.5% variation in rental can be explained by the variation in house value. This percentage is good enough to use the obtained regression equation to predict the rental income with help of value of houses.

 

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

 

I used excel data analysis tool to calculate the correlation coefficient between health expenditure and parental care.

 

According to the above output, Pearson correlation coefficient r = 0.1715

It looks like a weak positive linear relation exist between health expenditure and parental care.

Coefficient of determination r² = 0.1715² = 0.0294

So, only 2.94% variation in parental care can be explained by the health expenditure. So as the proportion is very low, it is not good to use the regression equation to predict parental care with help of health expenditure.

 

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

 

Hypothesis for correlation is positive:

Null Hypothesis: H₀: ? = 0

Alternate Hypothesis: H₁: ?> 0 (claim)

Level of significance? = 0.05

Excel output for regression analysis:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.764716
R Square	0.58479
Adjusted R Square	0.575764
Standard Error	1441.625
Observations	48
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1.35E+08	1.35E+08	64.78736	2.5E-10
Residual	46	95600982	2078282
Total	47	2.3E+08
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	5363.865	567.2408	9.456062	2.34E-12	4222.068	6505.661	4222.068	6505.661
Value	0.024358	0.003026	8.04906	2.5E-10	0.018267	0.03045	0.018267	0.03045

 

T test = 8.04906

P = 0 (approximately)

P < 0>

 

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

 

 

Null Hypothesis: H₀: ? = 0

Alternate Hypothesis: H₁: ?? 0 (claim)

Level of significance = 0.05

Excel output for regression analysis:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.171505
R Square	0.029414
Adjusted R Square	-0.04525
Standard Error	19.92675
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	1	156.4362	156.4362	0.393971	0.541089
Residual	13	5161.981	397.0755
Total	14	5318.417
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	69.7394	17.00601	4.100869	0.001251	33.00015	106.4786	33.00015	106.4786
Health Expenditure (% of GDP)	1.660599	2.645652	0.627671	0.541089	-4.05498	7.376182	-4.05498	7.376182

 

According to above output T test = 0.6277

P = 0.5411

As P >0.05 (level of significance), we fail to reject the null hypothesis. So, insufficient evidence to support the claim that there is a significant correlation between percentage spent on health expenditure and the percentage of women receiving prenatal care. 

 

 

 

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

 

Null Hypothesis: the activity and the time period are independent of each other for dolphins.

Alternate Hypothesis: the activity and the time period are dependent on each other for dolphins.

Level of significance ?= 0.01

Expected value table
	Period				Row
Activity	Morning	Noon	Afternoon	Evening	Total
Travel	14.857	3.095	4.746	16.302	39
Feed	33.524	6.984	10.709	36.783	88
Social	23.619	4.921	7.545	25.915	62
Column Total	72	15	23	79	189

 

Chi-Square test:

 

Statistic	DF	Value	P-value
Chi-square	6	68.464567	<0>

 

Chi square = 68.465

P = 0

As P <0>

Hence, sufficient evidence to support the claim that the activity and the time period are dependent on each other for dolphins.

 

 

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

 

Null Hypothesis: educational attainment and age are independent of each other

Alternate Hypothesis: educational attainment and age are dependent on each other

Level of significance ?= 0.05

Expected value
	Age Group					Row Total
Education	25-34	35-44	45-54	55-64	>64
Did not complete HS	11541.05	5762.05	6389.19	6330.01	7552.69	37575
Competed HS	13537.20	6758.66	7494.27	7424.86	8859.01	44074
College 1-3 years	6843.85	3416.90	3788.80	3753.70	4478.75	22282
College 4 or more years	8250.89	4119.39	4567.74	4525.43	5399.55	26863
Column Total	40173	20057	22240	22034	26290	130794

 

Chi-Square test:

 

Statistic	DF	Value	P-value
Chi-square	12	9.9325602e-16	1

 

P value = 1

As P > 0.05, we fail to reject the null hypothesis. So, insufficient evidence to support the claim that educational attainment and age are dependent on each other.

 

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

 

Null Hypothesis: The death from cardiovascular disease are in similar proportion as all deaths for the different age groups

Alternate Hypothesis: the death from cardiovascular disease are not in similar proportion as all deaths for the different age groups

Level of significance = 0.05

Age	Observed Frequency (O)	Proportion	Expected Frequency (E)	(O-E)^2/E
5   -   14	8	0.1	51.3	36.55
15 - 29	16	0.12	61.56	33.72
30  - 49	56	0.26	133.38	44.89
50  - 69	433	0.52	266.76	103.60
total	513		total	218.76

 

Chi square = 218.76

P = 0

As P < level>

 

 

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.

Null Hypothesis: Reason of choosing car is equally likely for all reasons

Alternate Hypothesis: Reason of choosing car is not equally likely for all reasons

Level of significance = 0.05

Reason	Observed Frequency (O)	Expected Frequency (E)	(O-E)^2/E
Safety	84	50	23.12
Reliability	62	50	2.88
cost	46	50	0.32
Performance	34	50	5.12
Comfort	47	50	0.18
Looks	27	50	10.58
Total	300	Sum	42.2

 

Chi square = sum of(O-E)^2/E =42.2

DF = 6 – 1 = 5

P = 0

As P < level>

 

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