Question

Bone mineral density and cola consumption has been recorded for a sample of patients. Let xx represent the number of colas consumed per week and yy the bone mineral density in grams per cubic centimeter. Based on the data shown below answer the questions rounding your final answers to four decimal places.

(a) Create a scatter plot with linear regression line for the data. (2 points)

y =  x +

(b) Interpret the slope of the regression equation in a complete sentence. (2 points)



(c) Use the linear correlation coefficient to determine if there is correlation. (5 points)

r =

Is there correlation at the 0.05 level of significance? (2 points)

• no
• yes

(d) According to the linear regression equation, the bone density of someone who drinks 23 colas per week is . (2 points)

Submit a file with your scatter plot below. (5 points)

Choose File No file chosen

x	y
2	1.0259
3	1.02021
4	1.01252
5	1.01683
6	1.00814
7	0.99845
8	1.00076
9	1.00607
10	1.00238
11	0.99469
12	1

Answer 1

Given,

Bone mineral density and cola consumption has been recorded for a sample of patients.

x: represent the number of colas consumed per week

y: The bone mineral density in grams per cubic centimetre.

The regression output summary for given data is as below

SUMMARY OUTPUT
­­
Regression Statistics
Multiple R	0.882578
R Square	0.778944
Adjusted R Square	0.754383
Standard Error	0.004925
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.000769	0.000769	31.71373	0.000321
Residual	9	0.000218	2.43E-05
Total	10	0.000988
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.026325	0.003607	284.5324	4.16E-19	1.018166	1.034485	1.018166	1.034485
x	-0.00264	0.00047	-5.63149	0.000321	-0.00371	-0.00158	-0.00371	-0.00158

(a) Create a scatter plot with linear regression line for the data.

--> Below is the scatter plot with the line of best fit/regression line

y = 1.026325 - 0.00264 * X

(b) Interpret the slope of the regression equation in a complete sentence. (2 points)

---> Interpretation of slope – If consumption of number of colas consumed per week is increased by 1, then bone mineral density changes by - 0.00264 amount.

(c) Use the linear correlation coefficient to determine if there is correlation. (5 points)

Linear correlation coefficient: r = -0.882578249

Is there correlation at the 0.05 level of significance?

----> The sample size is n = 11

The number of degrees of freedom is df = n-2 = 11 - 2 = 9

The corresponding critical correlation value rc for a significance level of α=0.05

rc=0.602

|r| > rc = 0.602 > rc hence the correlation coefficient is significant.

Answer --> yes

(d) According to the linear regression equation, the bone density of someone who drinks 23 colas per week is .

Given, X =23 and Since, y = 1.026325 - 0.00264 * X

y = 1.026325 - 0.00264 * 23 = 0.965605

at x=23 y = 0.965605