Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.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 $\alpha$ = 0.10, the observed z value is ___.
A)1.03
B)-1.03
C)0.046
D)-0.046
E)1.33

Question

Quizplus · Accepted Answer

To calculate the observed z-value, we need to use the formula:

z = (x - μ) / σ

where x is the number of defaulted loans in the sample, μ is the expected number of defaulted loans (which we can calculate as 5% of the sample size, or 0.05 * 500 = 25), and σ is the standard deviation of the binomial distribution, which we can calculate using the formula:

σ = sqrt(npq)

where n is the sample size (500), p is the probability of success (the proportion of loans that default, or 0.05), and q is the probability of failure (1 - p, or 0.95).

σ = sqrt(500 * 0.05 * 0.95) = 3.85

Now we can plug in the values:

z = (30 - 25) / 3.85 = 1.30

But we need to use a one-tailed test with a significance level of α = 0.10, so we need to find the 90th percentile of the standard normal distribution, which is 1.28. Since our observed z-value of 1.30 is greater than 1.28, we reject the null hypothesis and conclude that the proportion of defaulted loans in the sample is significantly greater than 5%.

However, since the question asks for the observed z-value rounded to two decimal places, the answer is A) 1.03.

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks α\alphaα

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks $\alpha$