Ophelia McPhee, VP of Consumer Credit, Confederation First Bank (CFB), monitors the default rate on personal loans among CFB members.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Using $\alpha$ = 0.01, the appropriate decision is ___.
A)reject the null hypothesis p 0.05
B)do not reject the null hypothesis p = 0.05
C)reject the null hypothesis $\mu$ 30
D)do not reject the null hypothesis $\mu$$\le$30
E)do nothing

Question

Ophelia McPhee, VP of Consumer Credit, Confederation First Bank (CFB), monitors the default rate on personal loans among CFB members.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Using $\alpha$ = 0.01, the appropriate decision is ___.
A)reject the null hypothesis p > 0.05
B)do not reject the null hypothesis p = 0.05
C)reject the null hypothesis $\mu$ > 30
D)do not reject the null hypothesis $\mu$$\le$30
E)do nothing

Quizplus · Accepted Answer

To determine the appropriate decision, we need to conduct a hypothesis test.

Null hypothesis: p ≤ 0.05 (default rate is no more than 5%)
Alternative hypothesis: p > 0.05 (default rate is more than 5%)

We can use a one-tailed Z-test for proportions since we have a large sample size (n=500) and can assume that the sampling distribution is approximately normal.

The test statistic is calculated as:

Z = (p̂ - p0) / sqrt(p0 * (1 - p0) / n)

where p̂ is the sample proportion of defaults (30/500 = 0.06), p0 is the hypothesized proportion (0.05), and n is the sample size (500).

Plugging in the values, we get:

Z = (0.06 - 0.05) / sqrt(0.05 * 0.95 / 500) = 1.41

Using a significance level of α = 0.01, the critical value for the Z-test is 2.33 (one-tailed test). Since our test statistic (1.41) is less than the critical value (2.33), we fail to reject the null hypothesis.

Therefore, the appropriate decision is to not reject the null hypothesis that the default rate is no more than 5%. Choice B is the correct answer.

Ophelia McPhee, VP of Consumer Credit, Confederation First Bank (CFB) α\alphaα

Ophelia McPhee, VP of Consumer Credit, Confederation First Bank (CFB) $\alpha$