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%	283
Humanities	25%	215
Natural Science	20%	220
Social Science	15%	102
All other subjects	8%	68

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 different.

H₁: The distributions are different.   

H₀: The distributions are different.

H₁: The distributions are the same.

H₀: The distributions are the same.

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?

chi-square

uniform    

binomial

normal

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

a)

level of significance =0.05

Ho: the distributions are the same

H1: the distributions are different

b)applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.320	283.000	284.160	-0.07	0.005
2	0.250	215.000	222.000	-0.47	0.221
3	0.200	220.000	177.600	3.18	10.123
4	0.150	102.000	133.200	-2.70	7.308
5	0.080	68.000	71.040	-0.36	0.130
total	1.000	888	888		17.7862

value of the chi-square statistic =17.786
Are all the expected frequencies greater than 5? :Yes
What sampling distribution will you use? chi-square
degrees of freedom =categories-1=4

c)

P-value < 0.005

d)

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

e)

At the 5% level of significance, the evidence is  sufficient to conclude ….