Question

A loan officer compares the interest rates for 48-month fixed-rate auto loans and 48-month variable-rate auto loans. Two independent, random samples of auto loan rates are selected. A sample of eight 48-month fixed-rate auto loans and a sample of five variable-rate auto loans had the following loan rates:

Fixed(%)	Variable(%)
4.29	3.59
3.75	2.75
3.5	2.99
3.99	2.5
3.75	3
3.99
5.4
4

Answer the following questions:

Let's define μFμF as the population mean loan rate for fixed-rate auto loans and

μVμV as the population mean loan rate for variable-rate auto loans.

a. The loan officer thinks the mean loan rates for fixed-rate auto loans is at least 1% greater than the mean loan rates for the variable-rate auto loans. Set up the null and alternative hypotheses needed to determine whether this claim is correct.

H0: μFμF-μVμV (Click to select)≠0<1≠1≤ 1=0≥1>1>0≥0≤ 0=1<0

Ha: μFμF-μVμV (Click to select)≤1>1≠0=0≤0>0 ≠1<0<1≥1=1

b. What is the critical value rejection rule. (Assume unequal variances and a significance level of 0.05.Answers should be in 4 decimals.)

Critical value rule: Reject H0 if (Click to select)t<-t alphat > t alphat > t alpha/2 or t <-t alpha/2 where the critical value is .

c. What is the t test statistic and p-value? (4 decimals)

t=

p-value=

What is the value in the numerator of the test statistic?

What is the value in the denominator of the test statistic?

Which one of the following is the correct p-value calculation?(Click to select)2 x (1-t.inv(0.95,10))1-t.dist(0.430,10,true)t.dist(0.430,10,true)2 x (1-t.dist(0.430,10,true))t.inv(0.95,10)1-t.inv(0.95,10)

d. With 95% confidence we (Click to select)cancannot  conclude that the mean loan rate for fixed-rate auto-loans is at least 1% greater than the loan rate for variable-rate auto-loans.

e. We have (Click to select)extremely strongsomenovery strongstrong evidence that the mean loan rates for fixed-rate auto loans is at least 1% greater than the mean loan rates for the variable rate auto loans.

f. Calculate a 95 percent confidence interval for the difference between the mean rates for fixed-rate and variable-rate 48-month auto loans.

_____ ≤  μFμF-μVμV  ≤ _____ (4 decimals).

Answer 1

a. Let X= the fixed rate auto loans. So we can say, where is unkown.

Let Y= the variable rate auto loans. So we can say, where is unkown.

So our null hypothesis can be stated as

against the alternate hypothesis,     

b. The critical value rule is given by : reject Ho if t_observed>t_a;10 (since it is a one sided test) where a=0.005

c. The test statistic is given by under Ho as

     

= 4.08

   The p-value is given by under Ho as

=  0.001

The value in the numerator of the test statistic = 1.118

The value in the denominator of the test statistic = 0.2740

d. With the 95% confidence interval, we can conclude that the mean loan rate for the fixed rate auto loans is at least 1% greater than variable rate auto loan because the tabulated value t_tabulated = 1.812 < t_observed = 4.08

e. We have extremely strong evidence that the mean loan rate for the fixed rate auto loans is at least 1% greater than variable rate auto loan because the p-value = 0.001 << 0.05 = a (the level of significance)

f. The required confidence interval is (1.3920 , -0.8439) using the formula where s.e. = standard error