Question

The Student Government Association at SLAY University wanted to demonstrate the relation ship between the number of beers a student drink and their blood alcohol content (BAC). A random sample of 18 students participated in a study in which each student was randomly assigned a number of cans of beers to drink. 30 minutes later their BAC was measured. Answer the following questions. Use alpha .05 for the test statistic.

	Beers	BAC
	6	.10
	3	.12
	4	.10
	7	.09
	5	.07
	6	.07
	3	.09
	4	.10
	2	.12
	2	.08
	3	.05
	4	.09
	1	.09
	3	.10
	4	.10
	2	.07
	2	.09
	5	.06

1. What is the coefficient of determination?                                                                                                                                                                  
2. What is the value of the correlation coefficients?                                                                                                                                                                   
3. How would you explain the value of this correlation coefficient to your boss?                                                                                                                                                            
4. What proportion of the variation in BAC is explained by variation in # beers?                  
5. What is the value of the calculated t (test statistic) for testing a hypothesis about the true correlation coefficient, rho?   
6. Can we concluded that the true correlation coefficient is not zero?                                                                                                                                                                
7. What is the estimated value of BAC for students that drink 8 beers?                                                                                                                                                               
8. Each additional beer increases a student's BAC by what amount?                                                                                                                                                                
9. Give an interpretation of the meaning of the intercepting the content of this problem.   
10. Is "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.
11. What is the null and alternative hypotheses?
12. PLEASE GIVE STEP BY STEP INSTRUCTIONS FOR ALL THE QUESTIONS AND EXPLAIN THE ANSWERS PLEASE

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
6	0.1	5.4444	0.0001	0.0272
3	0.12	0.4444	0.0010	-0.0211
4	0.1	0.1111	0.0001	0.0039
7	0.09	11.1111	0.0000	0.0056
5	0.07	1.7778	0.0003	-0.0244
6	0.07	5.4444	0.0003	-0.0428
3	0.09	0.4444	0.0000	-0.0011
4	0.1	0.1111	0.0001	0.0039
2	0.12	2.7778	0.0010	-0.0528
2	0.08	2.7778	0.0001	0.0139
3	0.05	0.4444	0.0015	0.0256
4	0.09	0.1111	0.0000	0.0006
1	0.09	7.1111	0.0000	-0.0044
3	0.1	0.4444	0.0001	-0.0078
4	0.1	0.1111	0.0001	0.0039
2	0.07	2.7778	0.0003	0.0306
2	0.09	2.7778	0.0000	-0.0028
5	0.06	1.7778	0.0008	-0.0378

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	66	1.59	46	0.00605	-0.08
mean	3.6667	0.0883	SSxx	SSyy	SSxy

sample size ,   n =   18      
here, x̅ =   3.666666667   ,   ȳ =   0.088333333
              
SSxx =    Σ(x-x̅)² =    46      
SSxy=   Σ(x-x̅)(y-ȳ) =   -0.08

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coefficient of determination,R² =    (Sxy)²/(Sx.Sy) =    0.0230

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correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.1516
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there is weak negative coorelation between the number of beers a student drink and their blood alcohol content (BAC)

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0.0230 proportion of the variation in BAC is explained by variation in # beers

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Ho:   ρ = 0  
Ha:   ρ ╪ 0  
n=   18  
alpha,α =    0.05  
correlation , r=   -0.1516  
t-test statistic =    t = r*√(n-2)/√(1-r²) =    -0.6137
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p-value =    0.5481

since,p-value>0.05, so do not reject H0

hence, we cannot concluded that the true correlation coefficient is not zero at α=0.05

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slope ,    ß1 = SSxy/SSxx =   -0.00174          
                  
intercept,   ß0 = y̅-ß1* x̄ =   0.09471          
                  
so, regression line is   Ŷ =   0.0947   +   -0.0017   *x
X=8

Ŷ =   0.0947   +   -0.0017   *8 = 0.081

estimated value of BAC for students that drink 8 beers=0.081
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Each additional beer increases a student's BAC by -0.0017

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when beer =0 then Students's BAC amount=0.0947

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slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   18              
alpha=   0.05              
estimated std error of slope =Se(ß1) =                s/√Sxx =    0.0028
                  
t stat =    ß1 /Se(ß1) =        -0.613684811      
                  
  
p-value =    0.5481              
decision :    p-value>α , do not reject Ho              

so,there is not enough evidence to conclude that beer is a useful predictor for BAC at α=0.05