Question

An advertising executive claims that there is a difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold. A random survey of 18 Visa Gold cardholders resulted in a mean household income of $73,480 with a standard deviation of $9200. A random survey of 9 MasterCard Gold cardholders resulted in a mean household income of $62,250 with a standard deviation of $10,300. Is there enough evidence to support the executive's claim? Let μ1 be the true mean household income for Visa Gold cardholders and μ2 be the true mean household income for MasterCard Gold cardholders. Use a significance level of α=0.1 for the test. Assume that the population variances are not equal and that the two populations are normally distributed.

Step 1 of 4: State the null and alternative hypotheses for the test.

Step 2 of 4: Compute the value of the t test statistic. Round your answer to three decimal places.

Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0H0. Round your answer to three decimal places.

Step 4 of 4: State the test's conclusion.

Answer 1

step 1 : Null hypothesis   

(i.e: null hypothesis states that there is no significant difference between true mean household income for Visa Gold cardholders and the  true mean household income for MasterCard Gold cardholders)

Alternative hypothesis is   

(i.e: alternative hypothesis states that there is significant difference between true mean household income for Visa Gold cardholders and the  true mean household income for MasterCard Gold cardholders)

step 2: The test statistic for the t test is

  

For the given problem

  

Thus the calculated t-statistic value is t = 2.763

steop 3:

at   and degrees of freedom 25 (n1+n2-2) the criitical value from t-distribution l table is 1.708

Thus the decision rule is we reject the null hypothesis if   

step 4: here the calculated value of test statistic t = 2.763

i.e; calculated t is greater than 1.708. Thus we reject the null hypothesis

Hence we conclude that  there is significant difference between true mean household income for Visa Gold cardholders and the  true mean household income for MasterCard Gold cardholders