Question

Because coles tend to replace healthier beverages and coles contain caffeine and phosphoric acid, researchers wanted to know whether cola consumption is sssociated with lower bone mineral densly in women. The accompanying data is the typical number of cans of cola consumed in a weak and the femoral nack bona mineral density for a sample of 15 women Complete paris (a) through (ti below Click the icon to view the woren's dala i Data Table colas per week 0 1. 1 1 (a) Find the Beet-equeres regression line realing cola consumption per weekes the explaralory variable. y=-.026x + (900 (Round to three dacimal placas as rended.) (6) Interpret the slope. Select the correct choice below and, if necessary, fill in the answer box to corrplete your choice. O A Far a bon density of Og/cm, the number of colas consumed is predicted to be {Round lo three decimal places as reeded.) OB. For every cola consumed per week, the hoor density decreases by gicman average {Round to three ducinal place as ruded.) OC. For every unit increase in bene density, the rumber of coas decreases by on average (Round to three decimal places as reeded.) OD. Far O colas consumed in a work, the bane tensity is predicted to begiam {Round to the decimal places as reeded.) E. It is not appropriate ta interarat the skape. (c) Interpret the y-intercept Select the correct choice below and, if necessary, ful in the answer box to complete your choice. O A For a bone density of Og/cm, the number of colas consumed la predicted to be in {Round to three decimal places as rended.) OB. For O colas consumed in a week, the bone density la predicted to be gcm {Round to the dacimal places as randnd.) OC. For every unit increase in bone density, the rumber of coas ciecreases by on average {Round to three decimal places as rended.) OD For every cola consumed per week, the bone denalty decreases by g/cm", on average. {Round to the decimal places as rended.) OE. I not appropriate to interpret the y-intercept 2 2 2 3 5 5 5 5 6 7 7 Bone Mineral Density (glom) B 0.907 0.882 0.882 0.871 0.854 0.839 0.836 0.819 0.766 0.770 0.766 0.765 0.737 0.721 (1.716 Print Done

Answer 1

a) -

Linear regression equation is as follows -

Y = a + bX

Where,

a = $\bar{y}$ - b

T

6 - Στη - α)(yi - g) Στη - 32

Observation table -

Sr. No.	Colas per week(x)	Bone mineral density(y)	(x-x_bar)	(y-y_bar)	(x-x_bar)(y-y_bar)	(x-x_bar)^2	(y-y_bar)^2
1	0	0.907	-3.47	0.1023	-0.3546	12.018	0.0105
2	1	0.889	-2.47	0.0843	-0.2079	6.0846	0.0071
3	1	0.802	-2.47	-0.0027	0.0067	6.0846	0
4	1	0.871	-2.47	0.0663	-0.1635	6.0846	0.0044
5	2	0.854	-1.47	0.0493	-0.0723	2.1512	0.0024
6	2	0.839	-1.47	0.0343	-0.0503	2.1512	0.0012
7	2	0.828	-1.47	0.0233	-0.0342	2.1512	0.0005
8	3	0.819	-0.47	0.0143	-0.0067	0.2178	0.0002
9	5	0.786	1.53	-0.0187	-0.0287	2.351	0.0003
10	5	0.77	1.53	-0.0347	-0.0532	2.351	0.0012
11	5	0.766	1.53	-0.0387	-0.0593	2.351	0.0015
12	5	0.765	1.53	-0.0397	-0.0609	2.351	0.0016
13	6	0.737	2.53	-0.0677	-0.1715	6.4176	0.0046
14	7	0.721	3.53	-0.0837	-0.2957	12.4842	0.007
15	7	0.716	3.53	-0.0887	-0.3134	12.4842	0.0079
total	52	12.07	-	-	-1.8655	77.7332	0.0504

Calculations -

$\bar{x} = \frac{\sum x_{i}}{n} = \frac{52}{15} = 3.4667 \approx 3.467$

$\bar{y} = \frac{\sum y_{i}}{n} = \frac{12.07}{15} = 0.8047 \approx 0.805$

$b = \frac{\sum (x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum (x_{i}-\bar{x})^{2} } = \frac{-1.8655}{77.7332} = -0.024$

a = $\bar{y}$ - b
T
= 0.805 - (-0.024)(3.467) = 0.805 + 0.083  = 0.8882 $\approx$ 0.888

Hence, linear regression is as follows -

Y = 0.888 - 0.024X

b) -

Interpretation of slope -

For every cola consumed per week, the bone density decreases by -0.024 g/cm³ on average.

Option b is correct.

c) -

Interpretation of Y-intercept -

For 0 colas consumed per week, the bone density is predicted to be 0.888 g/cm³.  

d) -

Bone density for 4 colas per week -

Y = 0.888 - 0.024X = 0.888 - 0.024(4) = 0.888 - 0.096 = 0.792

Predicted value of bone density for 4 colas per week is 0.792 g/cm³.

e) -

The researcher found woman who consumed 4 colas per week to have bone density of 0.786 g/cm³.

We have predicted that bone density for 4 colas per week is 0.792 g/cm³.

So, This women's bone density is below the average of 0.792 g/cm³.

f) -

Bone density for 2 colas per week -

Y = 0.888 - 0.024X = 0.888 - 0.024(2) = 0.888 - 0.048 = 0.84

Predicted value of bone density for 4 colas per week is 0.84 g/cm³.

So,it shows that predicted value for 2 colas consumed per week is within the scope of the model.

Hence, Option D is correct.

I.e. Yes, an X-value that represents a woman consuming 2 colas per week is possible & within the scope of the model.