Question

The data below are the final exam scores of 5 randomly selected calculus students and the number of hours they slept the night before the exam.

Hours, x	4	6	3	9	3
Scores, y	74	89	69	90	75

a) Draw scatterplot for the data.

b) Calculate the linear correlation coefficient to 3 decimal places.

(if you are unable to calculate the linear correlation coefficient, use .9 for part c,d and e)

c) Is there a linear relationship between the amount of sleep the student gets and the score on their exam. ( Justify your answer.Chart attached)

d) Determine the equation for the least squares regression line.

e) Predict the score of a student who studies 5 hours.

f) Interpret the slope and y-intercept of the least squares regression line

Answer 1

a)

b)

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
4	74	1	29.16	5.4
6	89	1	92.16	9.6
3	69	4	108.16	20.8
9	90	16	112.36	42.4
3	75	4	19.36	8.8

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	25	397	26	361.2	87
mean	5	79.4	SSxx	SSyy	SSxy

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.8978
c)

sample size ,   n =   5      
here, x̅ =   5       ȳ =   79.4
              
SSxx =    Σ(x-x̅)² =    26      
SSxy=   Σ(x-x̅)(y-ȳ) =   87      
              
slope ,    ß1 = SSxy/SSxx =   3.34615      

slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   5              
alpha=   0.05              
estimated std error of slope =Se(ß1) =                s/√Sxx =    0.9479
                  
t stat =    ß1 /Se(ß1) =        3.530058145      
                  
t-critical value=        3.182446305          
                  
p-value =    0.0386              
decision :    p-value<α , reject Ho              
so, there is linear relation the amount of sleep the student gets and the score on their exam

d)

slope ,    ß1 = SSxy/SSxx =   3.34615          
                  
intercept,   ß0 = y̅-ß1* x̄ =   62.66923          
                  
so, regression line is   Ŷ =   62.6692   +   3.3462   *x

e)

regression line is   Ŷ =   62.6692   +   3.3462   *x
x=5

predicted score , Ŷ =   62.6692   +   3.3462   *5=79.4

f)

slope

for every unit increase in amount of sleeps in hours X, the score y will get increase by 3.35

intercept-when amount of sleeps is 0 hours, then score y is 62.67