Tests of Independence An advertising agency is under contract to create a series
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Question
Tests of Independence An advertising agency is under contract to create a series of commercials for a soft drink company targeted to college students. To more effectively target the commercials, the agency wants to determine if soft drink preferences of college students depend on gender. They collect a random sample of 50 college students; the results are shown in the table below. 1. Gender Male | Femal101 AL marginal 23 13 14 50 Coke Soft Drink Sprite Mt. Dew 12 TOTAL25 a. Suppose a student is selected at random from this group. Find the probabilities of the following events: A: The person prefers Coke B: The person prefers Coke given that the person is a female (conditional) b. Are the events "female" and "prefers Coke" independent? Why or why not? Are Gender and Soft Drink associated? Perform a chi-square test. c. Gender Male Female TOTAL 23 13 14 50 Coke 15 Soft Drink | Sprite 5 Mt. Dew 12 TOTAL 25 25 Ho Ha : .Explanation / Answer
Solution:-
a).
A. Probability of a person prefers coke = 23/50 = 0.46
B. Probability that person prefers coke and is a female = 15/23 = 0.65
b). Are the events "Female" and "Prefers Coke" independent?
When two events are said to be independent of each other, this means that the probability that one event occurs in no way affects the probability of the other event occurring.
Mathematical representation,
P(Female and Prefer Coke) = P(Female) * P(Coke)
P(Female) = 25/50 = 0.50
P(Prefer Coke) = 23/50 = 0.46
As per the definition, P(Female and Coke) = 0.50 * 0.46 = 0.23
As per the given table, P(Female and Prefer coke) = 15/50 = 0.30
As the two values are different therfore, we can not conclude that the two are independent.
c). Are Gender and Soft drinks associated. Chi-square test.
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
H0: Gender and preferences are independent.
Ha: Gender and preferences are not independent.
Formulate an analysis plan. For this analysis, let the significance level be 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.
DF = (r - 1) * (c - 1) = (3 - 1) * (2 - 1) = 2
Er,c = (nr * nc) / n
E1,1 = (23 * 25) / 50 = 11.5
E1,2 = (23* 25) / 50 = 11.5
E2,1 = (13 * 25) / 50 = 6.5
E2,2 = (13 * 25) / 50 = 6.5
E3,1 = (14 * 25) / 50 = 7
E3,2 = (14 * 25) / 50 = 7
2 = [ (Or,c - Er,c)2 / Er,c ]
On substituting the values in above equation, we get X2 = 9.9656
The chi-square statistic is 9.9656.
where DF is the degrees of freedom, r is the number of levels of gender, c is the number of levels of the preference, nr is the number of observations from level r of gender, nc is the number of observations from level c of preference, n is the number of observations in the sample, Er,c is the expected frequency count when gender is level r and preference is level c, and Or,c is the observed frequency count when gender is level r preference is level c.
The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 9.9656.
We use the Chi-Square Distribution Calculator to find P(2 > 9.9656)
The p-value is 0.006855. The result is significant at p < .05.
Interpret results. Since the P-value (0.006855) is less than the significance level (0.05), we cannot accept the null hypothesis. Thus, we conclude that there is a relationship between gender and soft drink preference.
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