Question

The student government association wanted to demonstrate the relationship between the # of beers (“beers”) a student drinks and their blood alcohol content (“BAC”). A random sample of students participated in a study, and the data below was obtained. Use alpha = .05 for statistical tests.

Given the following two regression output tables, answer the questions:

ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.002811	0.002811	10.11611	0.015478078
Residual	7	0.001945	0.000278
Total	8	0.004756
	Coefficients	Standard Error	t Stat	P-Value
Intercept	0.04517045	0.020453362	2.20846114	0.06294
Beers	0.01198864	0.00376932	3.18058328	0.0154781

What is the coefficient of determination?

What is the value of the correlation coefficient?

How would you explain the value of this correlation coefficient to your boss?

What proportion of the variation in BAC is explained by variation in # of beers?

What is the value of the calculated t (test statistic) for testing a hypothesis about the true correlation coefficient, rho?

Can we conclude that the true correlation coefficient is not zero? (of course, given a reason why or why not)

What is the estimated value of BAC for a student that drinks 8 beers?

Each additional beer increases a student’s BAC by what amount?

Give an interpretation of the meaning of the intercept in the context of this problem.

Are “beers” a useful predictor for BAC? Give a brief reason why or why not, based on the data given, and NOT based on your personal experiences.

Please explain and give as much details as possible on how you get the answers. Please

Answer 1

coefficient of determination = SSR/SST = =0.002811/0.004756 = 0.5910

--------------------------------

value of the correlation coefficient,r = √R² = √0.5910 = 0.7688

---------------------------

there is moderate relationship between  the # of beers (“beers”) a student drinks and their blood alcohol content (“BAC”)

---------------------------

coefficient of determination = 0.5910

0.5910 of proportion of the variation in BAC is explained by variation in # of beers

----------------------

Ho:   ρ = 0                          
Ha:   ρ ╪ 0                          
                              
n=   9                          
alpha,α =    0.05                          
correlation , r =   0.7688                          
                              
Df =    n-2 =   7                      
t-test statistic = r*√(n-2)/√(1-r²) =        0.7688   * √    7   / √ ( 1 -    0.7688   ² ) =   3.180

----------------------------------

p-value =    0.0155   [excel function: =t.dist.2t(t-stat,df) ]       
decision:   p value < α , so, reject the null hypothesis          
So, we can conclude that the true correlation coefficient is not zero

-----------------------------------

estimated value of BAC for a student that drinks 8 beers is

=0.04514045+0.01198864*8= 0.14105

-------------------------------------------------

Each additional beer increases a student’s BAC by 0.01198864

------------------------

when # beers is zero, then students 's BAC is 0.04514045

---------------------------

p value for slope beer=0.015

p value <α=0.05, so, test is significant

So, beers” a useful predictor for BAC