Question

Mr. Acosta, a sociologist, is doing a study to see if there is a relationship between the age of a young adult (18 to 35 years old) and the type of movie preferred. A random sample of 93 adults revealed the following data. Test whether age and type of movie preferred are independent at the 0.05 level.

	Person's Age
Movie	18-23 yr	24-29 yr	30-35 yr	Row Total
Drama	7	16	11	34
Science Fiction	11	12	7	30
Comedy	9	7	13	29
Column Total	27	35	31	93

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Age and movie preference are not independent.
H₁: Age and movie preference are independent.H₀: Age and movie preference are independent.
H₁: Age and movie preference are not independent.    H₀: Age and movie preference are not independent.
H₁: Age and movie preference are not independent.H₀: Age and movie preference are independent.
H₁: Age and movie preference are independent.

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


Are all the expected frequencies greater than 5?

YesNo    

What sampling distribution will you use?

chi-squarenormal    binomialStudent's tuniform

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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, there is sufficient evidence to conclude that age of young adult and movie preference are not independent.At the 5% level of significance, there is insufficient evidence to conclude that age of young adult and movie preference are not independent.    

Answer 1

Given table data is as below

MATRIX	col1	col2	col3	TOTALS
row 1	7	16	11	34
row 2	11	12	7	30
row 3	9	7	13	29
TOTALS	27	35	31	N = 93

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calculation formula for E table matrix

E-TABLE	col1	col2	col3
row 1	row1*col1/N	row1*col2/N	row1*col3/N
row 2	row2*col1/N	row2*col2/N	row2*col3/N
row 3	row3*col1/N	row3*col2/N	row3*col3/N

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expected frequecies calculated by applying E - table matrix formulae

E-TABLE	col1	col2	col3
row 1	9.871	12.796	11.333
row 2	8.71	11.29	10
row 3	8.419	10.914	9.667

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calculate chisquare test statistic using given observed frequencies, calculated expected frequencies from above

Oi	Ei	Oi-Ei	(Oi-Ei)^2	(Oi-Ei)^2/Ei
7	9.871	-2.871	8.243	0.835
16	12.796	3.204	10.266	0.802
11	11.333	-0.333	0.111	0.01
11	8.71	2.29	5.244	0.602
12	11.29	0.71	0.504	0.045
7	10	-3	9	0.9
9	8.419	0.581	0.338	0.04
7	10.914	-3.914	15.319	1.404
13	9.667	3.333	11.109	1.149
				chisqr^2 o = 5.787

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set up null vs alternative as
null, Ho: age and movie preference are independent.
alternative, H1: age and movie preference are not independent.
level of significance, alpha = 0.05
from standard normal table, chi square value at right tailed, chisqr^2 alpha/2 =9.488
since our test is right tailed,reject Ho when chisqr^2 o > 9.488
we use test statistic chisqr^2 o = Σ(Oi-Ei)^2/Ei
from the table , chisqr^2 o = 5.787
critical value
the value of |chisqr^2 alpha| at los 0.05 with d.f (r-1)(c-1)= ( 3 -1 ) * ( 3 - 1 ) = 2 * 2 = 4 is 9.488
we got | chisqr^2| =5.787 & | chisqr^2 alpha | =9.488
make decision
hence value of | chisqr^2 o | < | chisqr^2 alpha | and here we do not reject Ho
chisqr^2 p_value =0.216
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(a) null, Ho: age and movie preference are independent.
alternative, H1: age and movie preference are not independent.
(b) test statistic: 5.787
   yes
   chi-square
(c) p-value:0.216, P-value > 0.100
critical value: 9.488
(d) since the P-value > α, we fail to reject the null hypothesis      
(e) at the 5% level of significance, there is insufficient evidence to conclude that age of young adult and movie preference are not independent