The bad debt ratio for a financial institution is defined to be the dollar value

Question

The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made. Suppose that a random sample of 7 Ohio banks is selected and that the bad debt ratios (written as percentages) for these banks are 8%, 7%, 9%, 7%, 7%, 5%, and 8%.

(a-1) Banking officials claim that the mean bad debt ratio for all Midwestern banks is 3.5 percent and that the mean bad debt ratio for Ohio banks is higher. Set up the null and alternative hypotheses needed to attempt to provide evidence supporting the claim that the mean bad debt ratio for Ohio banks exceeds 3.5 percent. (Round your answers to 1 decimal place. Omit the "%" sign in your response.)

H0: <  % versus Ha: >  %.

(a-2) Discuss the meanings of a Type I error and a Type II error in this situation.

(b) Assuming that bad debt ratios for Ohio banks are approximately normally distributed, use critical values and the given sample information to test the hypotheses you set up in part a by setting equal to .01. Also, interpret the p-value of 0.0001 for the test. (Round your answers to 3 decimal places.)

Since t.01 (Click to select)<> t, (Click to select)do not rejectreject H0 .

Type I: Conclude that Ohio’s mean bad debt ratio is (Click to select)> 3.5% when it actually is 3.5%. Type II: Conclude that Ohio’s mean bad debt ratio is (Click to select)> 3.5% when it actually is > 3.5%.

Explanation / Answer

Given that,
population mean(u)=3.5
sample data
(8,7,9,7,7,5,8)
caluclated sample mean, x =7.2857
caluclated standard deviation, s =1.2536
number (n)=7
null, Ho: <=3.5
alternate, H1: >3.5
level of significance, = 0.01
from standard normal table,right tailed t /2 =3.1427
since our test is right-tailed
reject Ho, if to > 3.1427
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =7.2857-3.5/(1.2536/sqrt(7))
to =7.99
| to | =7.99
critical value
the value of |t | with n-1 = 6 d.f is 3.1427
we got |to| =7.99 & | t | =3.1427
make decision
hence value of | to | > | t | and here we reject Ho
p-value :right tail - Ha : ( p > 7.9898 ) = 0.0001
hence value of p0.01 > 0.0001,here we reject Ho
ANSWERS
---------------
null, Ho: <=3.5
alternate, H1: >3.5
test statistic: 7.99
critical value: 3.1427
decision: reject Ho
p-value: 0.0001
we have evidence supporting the claim that the mean bad debt ratio for Ohio banks exceeds 3.5 percent.

Type I: Conclude that Ohio’s mean bad debt ratio is > 3.5% when it actually is <= 3.5%.
Type II: Conclude that Ohio’s mean bad debt ratio is <= 3.5% when it actually is > 3.5%.

Snce t.01 < t, reject HO

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