Answer:
See explanation
Step-by-step explanation:
Solution:-
- A survey was conducted among the College students for their motivations of using credit cards two years ago. A randomly selected group of sample size n = 425 college students were selected.
- The results of the survey test taken 2 years ago and recent study are as follows:
Old Survey ( % ) New survey ( Frequency )
Reward 27 112
Low rate 23 96
Cash back 21 109
Discount 9 48
Others 20 60
- We are to test the claim for any changes in the expected distribution.
We will state the hypothesis accordingly:
Null hypothesis: The expected distribution obtained 2 years ago for the motivation behind the use of credit cards are as follows: Rewards = 27% , Low rate = 23%, Cash back = 21%, Discount = 9%, Others = 20%
Alternate Hypothesis: Any changes observed in the expected distribution of proportion of reasons for the use of credit cards by college students.
( We are to test this claim - Ha )
We apply the chi-square test for independence.
- A chi-square test for independence compares two variables in a contingency table to see if they are related. In a more general sense, it tests to see whether distributions of categorical variables differ from each other.
- We will compute the chi-square test statistics ( X^2 ) according to the following formula:
![X^2 = Sum [ \frac{(O_i - E_i)^2}{Ei} ]](https://tex.z-dn.net/?f=X%5E2%20%3D%20Sum%20%5B%20%5Cfrac%7B%28O_i%20-%20E_i%29%5E2%7D%7BEi%7D%20%5D)
Where,
O_i : The observed value for ith data point
E_i : The expected value for ith data point.
- We have 5 data points.
So, Oi :Rewards = 27% , Low rate = 23%, Cash back = 21%, Discount = 9%, Others = 20% from a group of n = 425.
Ei : Rewards = 112 , Low rate = 96, Cash back = 109, Discount = 48, Others = 60.
Therefore,
![X^2 = [ \frac{(112 - 425*0.27)^2}{425*0.27} + \frac{(96 - 425*0.23)^2}{425*0.23} + \frac{(109 - 425*0.21)^2}{425*0.21} + \frac{(48 - 425*0.09)^2}{425*0.09} + \frac{(60 - 425*0.20)^2}{425*0.20}]\\\\X^2 = [ 0.06590 + 0.03132 + 4.37044 + 2.48529 + 7.35294]\\\\X^2 = 14.30589](https://tex.z-dn.net/?f=X%5E2%20%3D%20%5B%20%5Cfrac%7B%28112%20-%20425%2A0.27%29%5E2%7D%7B425%2A0.27%7D%20%2B%20%20%5Cfrac%7B%2896%20-%20425%2A0.23%29%5E2%7D%7B425%2A0.23%7D%20%2B%20%20%5Cfrac%7B%28109%20-%20425%2A0.21%29%5E2%7D%7B425%2A0.21%7D%20%2B%20%20%5Cfrac%7B%2848%20-%20425%2A0.09%29%5E2%7D%7B425%2A0.09%7D%20%2B%20%20%5Cfrac%7B%2860%20-%20425%2A0.20%29%5E2%7D%7B425%2A0.20%7D%5D%5C%5C%5C%5CX%5E2%20%3D%20%5B%200.06590%20%2B%200.03132%20%2B%204.37044%20%2B%20%202.48529%20%2B%20%207.35294%5D%5C%5C%5C%5CX%5E2%20%3D%2014.30589)
- Then we determine the chi-square critical value ( X^2- critical ). The two parameters for evaluating the X^2- critical are:
Significance Level ( α ) = 0.10
Degree of freedom ( v ) = Data points - 1 = 5 - 1 = 4
Therefore,
X^2-critical = X^2_α,v = X^2_0.1,4
X^2-critical = 7.779
- We see that X^2 test value = 14.30589 is greater than the X^2-critical value = 7.779. The test statistics value lies in the rejection region. Hence, the Null hypothesis is rejected.
Conclusion:-
This provides us enough evidence to conclude that there as been a change in the claimed/expected distribution of the motivations of college students to use credit cards.
