Question

The Nellie Mae organization found random sample of 100 students taken in 2004, average credit card balance = $2169. Suppose sample standard deviation of the credit card balances = $1000. Perform a test to test if the average credit card debt exceeds $2000.

a) what is the population, variable, parameter

b) does CLT hold

c) state hypothesis, H0 = , HA :   

d) calculate test stat

e) find p-value range

f) make decision at .05 significance level and write conclusion.

Answer 1

One-Sample t-test

The sample mean is Xˉ=2169, the sample standard deviation is s=1000, and the sample size is n=100. (1) Population - Population of students in 2004 Variable - the credit card balance for a student in 2004 Parameter - population average credit card balance for students in 2004 (2) CLT holds since our sample size = 100 is large enough (3) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =2000 Ha: μ >2000 This corresponds to a Right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. Critical Value Based on the information provided, the significance level is α=0.05, and the degree of freedom is n-1=100-1=99. Therefore the critical value for this Right-tailed test is tc​=1.6604. This can be found by either using excel or the t distribution table. Rejection Region The rejection region for this Right-tailed test is t>1.6604 (4)Test Statistics The t-statistic is computed as follows: (5) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is 0.0471 (6) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=1.69 > tc​=1.6604, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0471, and since p=0.0471≤0.05, it is concluded that the null hypothesis is rejected. Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is greater than 2000, at the 0.05 significance level.

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