Question

A student wants to see if the weight of a paperback book and the number of pages in the book are related. She guesses that the weight of the book will be a predictor of the number of pages it has. She randomly sampled 15 books from the library, test her claim at a 0.05 level of significance.

weight in ounces	number of pages
41	83
15	102
38	94
7	75
34	128
14	70
26	94
18	90
20	89
37	114
34	77
41	111
2	31
4	70
41	127

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 0.05 the critical value is -2.16 and 2.16

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

Let  x=the weight of a paperback book and y=the number of pages in the book

The student guesses that the weight of the book (x) will be a predictor of the number of pages it has (y).

that is we want to estimate the regression line

where is the intercept, is the slope of the regression line and is the random error

First we calculate the following of the sample

n=15 is the number of observations

The sample means are

The sum of squares are

Now we can get the correlation coefficient as

ans: r=0.712

An estimate of the slope is

An estimate of the intercept is

ans: The equation y=a+bx is:

Let be the true correlation coefficient between X and Y.

The hypotheses are:

The test statistics is

ans: The test value is 3.651

The degrees of freedom of the t statistics is n-2=15-2=13

this is a 2 tailed test (The alternative hypothesis has "not equal to")

The p-value is

Using technology such as Excel function =T.DIST.2T(3.651,13), we get the p-value=0.003

ans: The p-value=0.003

Critical value approach: We have been given that for α =0.05 the critical value is -2.16 and 2.16

that is we will reject the null hypothesis if the test statistics lies outside the range -2.16 to 2.16.

Here, The test statistics is 3.651 and it lies outside the range -2.16 to 2.16. Hence we will reject the null hypothesis

P-value approach: We will reject the null hypothesis if the p-value is less than the significance level α =0.05

Here, the p-value is 0.003 and it is less than α =0.05. Hence we reject the null hypothesis.

The decision is to

ans:

Thus the final conclusion sentence is

ans: There is enough evidence to support the claim that there is a linear relationship.