Question

The director of library services at a college did a survey of types of books (by subject) in the circulation library. Then she used library records to take a random sample of 888 books checked out last term and classified the books in the sample by subject. The results are shown below.

Subject Area	Percent of Books on Subject in Circulation Library on This Subject	Number of Books in Sample on This Subject
Business	32%	277
Humanities	25%	220
Natural Science	20%	212
Social Science	15%	110
All other subjects	8%	69

Using a 5% level of significance, test the claim that the subject distribution of books in the library fits the distribution of books checked out by students. (a) What is the level of significance?


State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are the same. H₀: The distributions are the same.
H₁: The distributions are different.     H₀: The distributions are different.
H₁: The distributions are the same. H₀: The distributions are different.
H₁: The distributions are different.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

Yes No    

What sampling distribution will you use?

normal

uniform    

chi-square

binomial

Student's t

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, the evidence is sufficient to conclude that the subject distribution of books in the library is different from that of books checked out by students.

At the 5% level of significance, the evidence is insufficient to conclude that the subject distribution of books in the library is different from that of books checked out by students.    

Answer 1

The statistical software output for this problem is:

Chi-Square goodness-of-fit results:
Observed: Oi
Expected: Ei

N	DF	Chi-Square	P-value
888	4	10.960914	0.027

Observed	Expected
277	284.16
220	222
212	177.6
110	133.2
69	71.04

Hence,

a) Level of significance = 0.05

H0: The distributions are the same.
H1: The distributions are different.

b) Test statistic = 10.961

Yes

Chi - square

c) 0.025 < P-value < 0.050

d) Since the P-value ≤ α, we reject the null hypothesis.

e) At the 5% level of significance, the evidence is sufficient to conclude that the subject distribution of books in the library is different from that of books checked out by students.