Question

At a major credit card​ bank, the percentages of people who historically apply for the​ Silver, Gold and Platinum cards are 60 %​, 30 % and 10 % respectively. In a recent sample of customers responding to a​ promotion, of 200​ customers, 103 applied for​ Silver, 68 for Gold and 29 for Platinum. Is there evidence to suggest that the percentages for this promotion may be different from the historical​ proportions? ​a) What is the expected number of customers applying for each type of card in this sample if the historical proportions are still​ true? ​b) Compute the chi squared statistic. ​c) How many degrees of freedom does the chi squared statistic​ have?

Answer 1

To determine if the percentages for the credit card promotion may be different from the historical proportions, a chi-square goodness of fit test can be conducted. The expected number of customers for each card type is calculated by multiplying the total number of customers by the respective historical percentages. The chi-squared statistic is computed using the formula ((observed - expected)² / expected), and the degrees of freedom is (number of categories - 1).

Step-by-step explanation:

To determine if there is evidence to suggest that the percentages for this promotion may be different from the historical proportions, we can conduct a chi-square goodness of fit test.

a) The expected number of customers applying for each type of card in this sample, if the historical proportions are still true, can be calculated by multiplying the total number of customers (200) by the respective historical percentages: 200 * 0.6 = 120 for Silver, 200 * 0.3 = 60 for Gold, and 200 * 0.1 = 20 for Platinum.

b) The chi-squared statistic can be computed by using the formula: chi-squared = Σ((observed - expected)² / expected), where Σ is the sum of all the calculations for each category. In this case, the calculations would be ((103 - 120)² / 120) + ((68 - 60)² / 60) + ((29 - 20)^2 / 20).

c) The degrees of freedom for the chi-squared statistic in this case is (number of categories - 1), which would be (3 - 1) = 2.

Answer 2

a)
`E_(Gold) =0.3*200=60`

`E_(Silver) =0.6*200=120`

`E_(Platinum) =0.1*200=20`

b)
`\chi^2 = ((68-60)^2)/(60)+((103-120)^2)/(120)+((29-20)^2)/(20) =7.525`

c)
`df=Categories-1=3-1=2`

Step-by-step explanation:

A chi-square goodness of fit test "determines if a sample data matches a population".

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 another".

Assume the following dataset:

Silver, Gold and Platinum cards are 60 %​, 30 % and 10 %

We need to conduct a chi square test in order to check the following hypothesis:

H0: There is no difference with the proportions claimed

H1: There is a difference with the proportions claimed

The level of significance assumed for this case is
`\alpha=0.05`

The statistic to check the hypothesis is given by:

`\chi^2 =\sum_(i=1)^n ((O_i -E_i)^2)/(E_i)`

The observed values are:

`O_(Gold)=68`

`O_(Silver)=103`

`O_(Platinum)=29`

a) What is the expected number of customers applying for each type of card in this sample if the historical proportions are still​ true?

The expected values are given by:

`E_(Gold) =0.3*200=60`

`E_(Silver) =0.6*200=120`

`E_(Platinum) =0.1*200=20`

b) Compute the chi squared statistic. ​

And now we can calculate the statistic:

`\chi^2 = ((68-60)^2)/(60)+((103-120)^2)/(120)+((29-20)^2)/(20) =7.525`

Now we can calculate the degrees of freedom for the statistic given by:

`df=Categories-1=3-1=2`

And we can calculate the p value given by:

`p_v = P(\chi^2_(2) >7.525)=0.0232`

And we can find the p value using the following excel code:

"=1-CHISQ.DIST(7.525,2,TRUE)"

Since the p value is lower than the significance level we reject the null hypothesis at 5% of significance, and we can conclude that we have significant differences from the % assumed for each category.