Question

A student wants to see if the number of times a book has been checked out of the library in the past year and the number of pages in the book are related. She guesses that the number of times a book has been checked out of the library in the past year will be a predictor of the number of pages it has. She randomly sampled 15 books from the library, test her claim at a 0.01 level of significance.

number of times checked out	number of pages
3	322
8	277
13	272
19	248
50	424
46	249
50	354
42	254
28	372
19	249
9	392
26	274
14	375
29	314
30	317

The correlation coefficient:

r= (round to 3 decimal places)

The equation y=a+bx is: (round to 3 decimal places)

y=+ x

The hypotheses are:

H0:ρ=0H0:ρ=0 (no linear relationship)
HA:ρ≠0HA:ρ≠0 (linear relationship) (claim)

Since αα is 0.01 the critical value is -3.012 and 3.012

The test value is:  (round to 3 decimal places)

The p-value is:  (round to 3 decimal places)

The decision is to

• reject H0H0
• do not reject H0H0

Thus the final conclusion sentence is

• There is enough evidence to reject the claim that there is a linear relationship.
• There is not enough evidence to reject the claim that there is a linear relationship.
• There is enough evidence to support the claim that there is a linear relationship.
• There is not enough evidence to support the claim that there is a linear relationship.

Answer 1

ΣX = 386 ΣY=4693   ΣX * Y = 122117    ΣX2 = 13322 ΣY2 = 1516345

r = 0.106

Equation of regression line is Ŷ = a + bX


b = 0.398
a =( Σ Y - ( b * Σ X) ) / n
a =( 4693 - ( 0.3985 * 386 ) ) / 15
a = 302.612
Equation of regression line becomes Ŷ = 302.612 + 0.398 X

To Test :-

H0 :- ρ = 0
H1 :- ρ ≠ 0

Test Statistic :-
t = (r * √(n - 2) / (√(1 - r2))
t = ( 0.1058 * √(15 - 2) ) / (√(1 - 0.0112) )
t = 0.384

P - value = P ( t > 0.3836 ) = 0.707

Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.7075 > 0.01 ,hence we fail to reject null hypothesis
Conclusion :- We Accept H0

There is not enough evidence to support the claim that there is a linear relationship.