A loan officer compares the interest rates for 48-month fixed-rate auto loans an
ID: 3260730 • Letter: A
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 48month fixedrate auto loans had the following loan rates:
while a sample of five 48month variablerate auto loans had loan rates as follows:
Figure 10.7
(a) 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.
Hf – µa: µf – µv =
(b) Figure 10.7 gives the Excel output of using the equal variances procedure to test the hypotheses you set up in part a. Assuming that the normality and equal variances assumptions hold, use the Excel output and critical values to test these hypotheses by setting equal to .10, .05, .01, and .001. How much evidence is there that the mean rates for 48month fixed and variablerate auto loans differ? (Round your answer to 3 decimal places.)
(c) Figure 10.7 gives the p–value for testing the hypotheses you set up in part a. Use the p–value to test these hypotheses by setting equal to .10, .05, .01, and .001. How much evidence is there that the mean rates for 48month fixed and variablerate auto loans differ? (Round your answer to 4 decimal places.)
(d) Calculate a 95 percent confidence interval for the difference between the mean rates for fixed and variablerate 48month auto loans. Can we be 95 percent confident that the difference between these means exceeds .4 percent? (Round your answers to 3 decimal places. Negative value should be indicated by a minus sign.)
Confidence interval = [ , ]. (Click to select)NoYes, the entire interval is (Click to select)abovenot above .40.
(e) Use a hypothesis test to establish that the difference between the mean rates for fixed and variablerate 48month auto loans exceeds .4 percent. Use equal to .05. (Round your t answer to 3 decimal places and other answers to 1 decimal place.)
8.71% 7.13% 8.11% 8.87% 7.87% 8.29% 7.43% 7.86%Explanation / Answer
Answer:
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 48month fixedrate auto loans had the following loan rates:
8.71%
7.13%
8.11%
8.87%
7.87%
8.29%
7.43%
7.86%
while a sample of five 48month variablerate auto loans had loan rates as follows:
7.67%
6.52%
7.69%
6.56%
7.30%
Figure 10.7
Excel Output of Testing the Equality
of Mean Loan Rates for Fixed and Variable
48-Month Auto Loans
t-Test: Two-Sample Assuming Equal Variances
Fixed-Rate (%)
Variable-Rate (%)
Mean
8.03375
7.14800
Variance
.352341
.33237
Observations
8
5
Pooled Variance
.345079
Hypothesized Mean Difference
0
df
11
t Stat
2.644906
P(T<=t) one-tail
.011396
t Critical one-tail
1.79588
P(T<=t) two-tail
.022793
t Critical two-tail
2.200985
(a) 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.
H0: µf – µv = 0
Ha: µf – µv 0
(b) Figure 10.7 gives the Excel output of using the equal variances procedure to test the hypotheses you set up in part a. Assuming that the normality and equal variances assumptions hold, use the Excel output and critical values to test these hypotheses by setting equal to .10, .05, .01, and .001. How much evidence is there that the mean rates for 48month fixed and variablerate auto loans differ? (Round your answer to 3 decimal places.)
t = 2.645 with 11 df
Reject H0 at = 0.1, and 0.05 but not at =0.01 and 0.001
Strong evidence that rates differ.
(c) Figure 10.7 gives the p–value for testing the hypotheses you set up in part a. Use the p–value to test these hypotheses by setting equal to .10, .05, .01, and .001. How much evidence is there that the mean rates for 48month fixed and variablerate auto loans differ? (Round your answer to 4 decimal places.)
p–value = 0.0228
Reject H0 at = 0.05, and 0.1, but not at = 0.01 and 0.001
Strong evidence.
(d) Calculate a 95 percent confidence interval for the difference between the mean rates for fixed and variablerate 48month auto loans. Can we be 95 percent confident that the difference between these means exceeds .4 percent? (Round your answers to 3 decimal places. Negative value should be indicated by a minus sign.)
Confidence interval = [ 0.149, 1.623 ]. (No, the entire interval is not above .40.
(e) Use a hypothesis test to establish that the difference between the mean rates for fixed and variablerate 48month auto loans exceeds .4 percent. Use equal to .05. (Round your t answer to 3 decimal places and other answers to 1 decimal place.)
H0: µf – µv =0.4 versus Ha: µf – µv 0.4
t = 1.450
Do not reject H0 with a = .05.
Hypothesis Test: Independent Groups (t-test, pooled variance)
f
v
8.0338
7.1480
mean
0.5936
0.5765
std. dev.
8
5
n
11
df
0.88575
difference (f - v)
0.34508
pooled variance
0.58743
pooled std. dev.
0.33489
standard error of difference
0.4
hypothesized difference
1.450
t
.0874
p-value (one-tailed, upper)
0.14866
confidence interval 95.% lower
1.62284
confidence interval 95.% upper
0.73709
margin of error
8.71%
7.13%
8.11%
8.87%
7.87%
8.29%
7.43%
7.86%
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