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	6	17	11	34
Science Fiction	10	9	11	30
Comedy	7	10	12	29
Column Total	23	36	34	93

(a) What is the level of significance?


State the null and alternate hypotheses.

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

B) H₀: Age and movie preference are independent.
H₁: Age and movie preference are independent.   

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

D) H₀: Age and movie preference are not 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?

A) Yes

B) No     

What sampling distribution will you use?

A) Student's t

B) normal    

C) uniformchi-square

D) binomial

What are the degrees of freedom?


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

A) P-value > 0.100

B) 0.050 < P-value < 0.100    

C) 0.025 < P-value < 0.050

D) 0.010 < P-value < 0.025

E) 0.005 < P-value < 0.010

F) 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?

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

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

C) 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.

A) At the 5% level of significance, there is sufficient evidence to conclude that age of young adult and movie preference are not independent.

B) 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

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

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

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

b)

Yes all the expected frequencies greater than 5.  

C) The sampling distribution will we use uniformchi-square.

The degrees of freedom is

DF = (r - 1) * (c - 1) = (3 - 1) * (3 - 1)
D.F = 4
E_r,c = (n_r * n_c) / n


Χ² = 3.78

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 4 degrees of freedom is more extreme than 3.78.

c) We use the Chi-Square Distribution Calculator to find P(Χ² > 3.78) = 0.437

A) P-value > 0.100

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

Interpret results. Since the P-value (0.437) is greater than the significance level (0.05), we have to accept the null hypothesis. Thus, we conclude that there is a relationship between Age and movie preference.

e) B) At the 5% level of significance, there is insufficient evidence to conclude that age of young adult and movie preference are not independent.