Question

NOTE: I am just checking my own work so if you respond, please make sure I can read your handwriting.

A loan officer wants to compare the interest rates for 48-month fixed-rate auto loans and 48-month variable-rate auto loans.

Two independent, random samples of auto loans are selected. A sample of eight 48-month fixed-rate auto-loans have the following loan rates:

4.29%, 3.75%, 3.50%, 3.99%, 3.75%, 3.99%, 5.40%, 4.00%

while a sample of five 48-month variable-rate auto loans have the loan rates as follows:

3.59%, 2.75%, 2.99%, 2.50%, 3.00%

Test statistic and the corresponding p value s are listed below:

Test Statistic	p Value
3.7431	0.0032

Tasks:

• Set up the null and alternative hypotheses needed to determine whether the mean rates for 48-month fixed-rate and variable-rate auto loans differ.
• Identify the test you will apply to test the hypothesis. Justify your choice.
• Choose an appropriate level of significance.
• Define type I and type II errors in the context of your hypotheses.
• State your decision regarding the hypothesis.
• State the conclusion.

Answer 1

Answer:

Null Hypothesis, H₀: Mean rates for 48-month fixed-rate and variable-rate auto loans are equal.

Alternate Hypothesis, Ha: Mean rates for 48-month fixed-rate and variable-rate auto loans are not equal.

The Excel output of 2-sample t-test assuming the equal variances procedure to test the hypotheses is shown below:

We use this test because the sample sizes are small and the pooled variance t-test is an appropriate test in such cases.

An appropriate level of significance is 0.95,alpha =0.05

We can see from the above that t-stat at 3.7431 > the t-critical (2-tail) 2.2. Also, the p-value 0.003 <0.05. Hence it lies in the rejection region and we reject the Null Hypothesis H₀.

A type I error would occur if the Null Hypothesis was true, but we rejected the Null hypothesis due to wrong outcome of the test. In our example, this would mean the Mean rates for 48-month fixed-rate and variable-rate auto loans are equal, but we conclude they are not equal by rejecting the Null Hypothesis.

A type II error would occur if the Null Hypothesis was false, but we failed to reject the Null Hypothesis due to wrong outcome of the test. In our case, this would mean the Mean rates for 48-month fixed-rate and variable-rate auto loans are NOT equal, but we conclude they are equal by not rejecting the Null Hypothesis..

Decision regarding the hypothesis: we reject the Null Hypothesis H₀.

Conclusion: The Mean rates for 48-month fixed-rate and variable-rate auto loans are NOT equal at 0.95 levelof significance.