Question

A survey was conducted two years ago asking college students their top motivations for using a credit card. To determine whether this distribution has​ changed, you randomly select 425 college students and ask each one what the top motivation is for using a credit card. Can you conclude that there has been a change in the claimed or expected​ distribution? Use alpha = 0.10.

Response Old Survey % New Survey Frequency
Reward 27 112
Low rate 23 96
Cash back 21 109
Discount 9 48
Others 20 60

Answer 1

See explanation

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 [ ((O_i - E_i)^2)/(Ei) ]`

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 = [ ((112 - 425*0.27)^2)/(425*0.27) + ((96 - 425*0.23)^2)/(425*0.23) + ((109 - 425*0.21)^2)/(425*0.21) + ((48 - 425*0.09)^2)/(425*0.09) + ((60 - 425*0.20)^2)/(425*0.20)]\\\\X^2 = [ 0.06590 + 0.03132 + 4.37044 + 2.48529 + 7.35294]\\\\X^2 = 14.30589`

- 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.