mortgage broker is offering home mortgages at a rate of 9.5%, but the broker is fearful that this value

is higher than many others are charging. A sample of 40 mortgages filed in the county courthouse shows an
average of 9.25% with a standard deviation of 8.61%. Does this sample indicate a smaller average? Use α
= 0.05 and assume a normally distributed population.
A. No, because the test statistic falls in the acceptance region.
B. Yes, because the test statistic is greater than –1.645.
C. Yes, because the sample mean of 9.25 is below 9.5.
D. No, because the test statistic is –1.85 and falls in the rejection region

im not sure im having trouble with this myself. Was anyone able to find the answer?

To determine if the sample average of 9.25% is significantly smaller than the mortgage broker's offer of 9.5%, we can conduct a hypothesis test using the information given.

Step 1: State the hypothesis:
The null hypothesis (H0) states that there is no significant difference between the sample mean and the broker's offer: μ = 9.5%.
The alternative hypothesis (Ha) states that the sample mean is smaller than the broker's offer: μ < 9.5%.

Step 2: Set the significance level (α):
The significance level (α) is given as 0.05, which means we are willing to accept a 5% chance of making a Type I error.

Step 3: Calculate the test statistic:
To calculate the test statistic, we can use the formula:
test statistic (t) = (sample mean - hypothesized mean) / (standard deviation / √sample size)

In this case:
sample mean = 9.25%
hypothesized mean = 9.5%
standard deviation = 8.61%
sample size = 40

Plugging these values into the formula, we get:
t = (9.25% - 9.5%) / (8.61% / √40)

Step 4: Determine the critical region:
Since the alternative hypothesis is that the sample mean is smaller, we need to find the critical t-value for a one-tailed test with a significance level of 0.05.

Using a t-distribution table (or a statistical calculator), the critical t-value is approximately -1.684 (for a one-tailed test).

Step 5: Make a decision:
If the test statistic falls in the rejection region (beyond the critical t-value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the test statistic is calculated as -1.85, which is smaller than -1.684. Therefore, the test statistic falls in the rejection region.

Answer: D. No, because the test statistic is -1.85 and falls in the rejection region.