Question

The authors of a paper studied characters who were depicted smoking in movies released between a certain range of years. The smoking characters were classified according to sex and whether the character type was positive, negative, or neutral. The resulting data are given in the table. Assume that this sample is representative of smoking movie characters. Do these data provide evidence of an association between sex and character type for movie characters who smoke? Use α = 0.05.

	Character Type
Sex	Positive	Negative	Neutral
Male	255	107	131
Female	85	11	50

State the appropriate null and alternative hypotheses.

H₀: The proportions falling into the character type categories are not the same for the two sexes.
H_a: The proportions falling into the character type categories are the same for the two sexes.

H₀: There is no association between sex and character type.
H_a: There is an association between sex and character type.

H₀: The proportions falling into the character type categories are the same for the two sexes.
H_a: The proportions falling into the character type categories are not the same for the two sexes.

H₀: There is an association between sex and character type.
H_a: There is no association between sex and character type.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X²=

P-value=

State the conclusion in the problem context.

Fail to reject H₀. There is not convincing evidence to conclude that there is an association between sex and character type for movie characters who smoke.

Fail to reject H₀. There is not convincing evidence to conclude that the proportions falling into the character type categories are not the same for the two sexes.

Reject H₀. There is convincing evidence to conclude that there is an association between sex and character type for movie characters who smoke.

Reject H₀. There is convincing evidence to conclude that the proportions falling into the character type categories are not the same for the two sexes.

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: There is no association between sex and character type.

Alternative hypothesis: H_a: There is an association between sex and character type.

We are given level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 2

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 1*2 = 2

α = 0.05

Critical value = 5.991465

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	C1	C2	C3	Total
R1	255	107	131	493
R2	85	11	50	146
Total	340	118	181	639

Expected Frequencies
	Column variable
Row variable	C1	C2	C3	Total
R1	262.3161	91.03912	139.6448	493
R2	77.68388	26.96088	41.35524	146
Total	340	118	181	639

Calculations
(O - E)
-7.31612	15.96088	-8.64476
7.316119	-15.9609	8.644757

(O - E)^2/E
0.20405	2.798243	0.535157
0.689018	9.448861	1.80707

Chi square = ∑[(O – E)^2/E] = 15.4824

Chi square = 15.482

P-value = 0.0004

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that there is an association between sex and character type.

Reject H₀. There is convincing evidence to conclude that there is an association between sex and character type for movie characters who smoke.