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 10 books from the library, test her claim at a 0.01 level of significance.

number of times checked out	number of pages
4	277
31	303
15	242
12	382
25	329
22	263
37	361
43	323
44	278
31	350

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 -3.355 and 3.355

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

Solution :-

1). Thr correlation coefficient:

r= 0.182

To find this result run the following code in python.

  times_checked=[4,31,15,12,25,22,37,43,44,31]
pages=[277,303,242,382,329,263,361,323,278,350]

  from scipy.stats import pearsonr

  corr,_= pearsonr(times_checked, pages)

print (corr)

0.18231726381436067

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

y=294.258+ 0.627x

These values that are unerlied are naswer to this question and this we can obtain by running a linear regression. This result for linear regression can be obtained by running following code of python.

from sklearn.linear_model import LinearRegression

times_checked=np.array([4,31,15,12,25,22,37,43,44,31]).reshape(-1,1)
pages=np.array([277,303,242,382,329,263,361,323,278,350])

  model = LinearRegression().fit(times_checked, pages)

  print('intercept:', model.intercept_)
print('slope:', model.coef_)

  intercept: 294.257935516121

slope: [0.62659335]

3).The hupothesis to be tested in this case is as follows;-

Since

Test statistics

4). Now having found the answer for t-value as 0.524, we can find the p-value for this t-value. To find the p-value we need to do the following calculation:-

  

5). Hence the decision is do not  reject H0.

We reach to this decision because p-value is far greater than value of level of signifance i.e. 0.01.

We can also say that we fail to reject null hypothesis because t-statistics because t-statistics falls between the critical-region (-3.355,3.355).

6). Thus the final conclusion sentence is-

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

This is because we fail to reject Null hypothesis.