Question

1. Check that the measurements for the heart rate are coming from a normal distribution by constructing a histogram.
2. Create indicator variables for female and good AQI. Fit the regression model and discuss its goodness of fit. What portion of variation in heart rate can this model explain?
3. Write the fitted model. What variables are significant predictors of heart rate at the 5% level of significance?
4. Plot residuals against predicted values. Does it show any pattern?
5. Compute the predicted heart rate of a 50-year-old male who has a BMI of 20, is not taking any heart medications, and resides in an area with a good air quality

A cardiologist conducts a study to find out what factors are good predictors of elevated heart rate (HR) in her patients. She measures heart rate at rest in 30 patients on their next visit, and obtains from the medical charts additional information on their age, gender, body mass index (BMI), and the number of currently taken heart medications. She also obtains the air quality index (AQI) for area of residence of her patients (unhealthy or good). The data are given in the file

age	gender	BMI	nmeds	AQI	HR
69	M	24.1	2	unhealthy	94
54	M	29.6	0	unhealthy	92
57	F	20.2	2	good	81
71	F	21.5	2	good	100
62	M	27.4	3	good	79
58	M	18.9	2	good	79
65	F	22.2	1	good	106
70	F	25.9	1	unhealthy	117
67	F	23.4	1	good	94
63	M	23.8	2	unhealthy	108
55	M	24.6	0	good	94
64	F	31.4	3	good	97
63	M	28	2	good	91
66	M	22.9	2	good	86
44	F	17.2	0	unhealthy	86
53	M	25.2	0	good	84
71	F	20.3	2	unhealthy	111
49	M	17.1	1	good	75
65	F	23.4	2	unhealthy	114
45	F	19	2	unhealthy	83
56	F	22.9	3	unhealthy	112
74	M	32.4	1	good	97
48	F	29.9	0	good	76
50	F	23.9	1	unhealthy	97
66	F	27.8	3	good	82
73	F	24.8	3	good	105
61	M	32.8	1	good	84
82	M	29.7	0	good	92
72	F	25.2	0	good	114
59	F	22.6	0	good	86

Answer 1

Descriptive Statistics
	Mean	Std. Deviation	N
HR	93.8667	12.37833	30
Age	61.7333	9.32528	30
Gender	.4333	.50401	30
AQI	.6667	.47946	30
BMI	24.6293	4.26183	30

heart rate (HR) on average 93.8667 of  patients. On average age 61.7333 of  patients.

Correlations
	HR	Age	Gender	AQI	BMI
Pearson Correlation	HR	1.000	.505	-.361	-.438	.025
Age	.505	1.000	.047	.234	.360
Gender	-.361	.047	1.000	.190	.267
AQI	-.438	.234	.190	1.000	.268
BMI	.025	.360	.267	.268	1.000
Sig. (1-tailed)	HR	.	.002	.025	.008	.447
Age	.002	.	.402	.107	.025
Gender	.025	.402	.	.157	.077
AQI	.008	.107	.157	.	.076
BMI	.447	.025	.077	.076	.
N	HR	30	30	30	30	30
Age	30	30	30	30	30
Gender	30	30	30	30	30
AQI	30	30	30	30	30
BMI	30	30	30	30	30

Heart Rate to BMI and age as positive correlation.

Heart Rate to Gender and AQI are perfect positive correlation.

Model Summaryb
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate	Change Statistics	Durbin-Watson
R Square Change	F Change	df1	df2	Sig. F Change
1	.814a	.663	.609	7.73649	.663	12.310	4	25	.000	2.234
a. Predictors: (Constant), BMI, Gender, AQI, Age
b. Dependent Variable: HR

R_Squre = 0.663

66.3% indicates that the model explains all the variability of the HR data around its mean.It's mean's that the model is good fit for the heart rate data.

ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	2947.134	4	736.784	12.310	.000b
Residual	1496.332	25	59.853
Total	4443.467	29
a. Dependent Variable: HR
b. Predictors: (Constant), BMI, Gender, AQI, Age

Coefficientsa
Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.	95.0% Confidence Interval for B	Correlations
B	Std. Error	Beta	Lower Bound	Upper Bound	Zero-order	Partial	Part
1	(Constant)	52.661	11.034		4.773	.000	29.936	75.386
Age	.847	.168	.638	5.054	.000	.502	1.192	.505	.711	.587
Gender	-7.216	2.991	-.294	-2.413	.024	-13.377	-1.056	-.361	-.435	-.280
AQI	-13.833	3.177	-.536	-4.354	.000	-20.377	-7.289	-.438	-.657	-.505
BMI	.052	.380	.018	.136	.893	-.731	.834	.025	.027	.016
a. Dependent Variable: HR Regression model : HR = 53.388 + 0.854 *Age - 7.119 *Gender - 13.763 * AQI The predicate heart rate 53.388 as per one unit of heart rate. The predicate heart rate to  age increases by 0.845 year as per one unit of heart rate. The predicate heart rate to  Gender decreases by -7.119  as per one unit of heart rate. The predicate heart rate to  AQI decreases by -13.763 year as per one unit of heart rate.

Coefficient Correlationsa
Model	BMI	Gender	AQI	Age
1	Correlations	BMI	1.000	-.240	-.161	-.325
Gender	-.240	1.000	-.138	.075
AQI	-.161	-.138	1.000	-.162
Age	-.325	.075	-.162	1.000
Covariances	BMI	.144	-.273	-.195	-.021
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