Net interest margin—often referred to as spread—is the difference between the rate banks pay on deposits and the rate they charg

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Net interest margin—often referred to as spread—is the difference between the rate banks pay on deposits and the rate they charg

e for loans. Suppose that the net interest margins for all U.S. banks are normally distributed with a mean of 4.15 percent and a standard deviation of .5 percent. (a) Find the probability that a randomly selected U.S. bank will have a net interest margin that exceeds 5.40 percent. (Round answer to 4 decimal places.) P (b) Find the probability that a randomly selected U.S. bank will have a net interest margin less than 4.40 percent. (Round answer to 4 decimal places.) P (c) A bank wants its net interest margin to be less than the net interest margins of 95 percent of all U.S. banks. Where should the bank’s net interest margin be set? (Round the z value to 3 decimal places. Round answer to 4 decimal places.)

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1 answer:

Nady [3.2K] · Answer 1 · 2026-04-01T10:45:06+03:00

Answer:

(a) $P(X\:>\:5.40)=0.9938$

(b) $P(X\:$

(c) X=4.975 percent

Explanation:

(a) Identify the z-value that represents 5.40 percent

. $Z=\frac{X-\mu}{\sigma}$

$Z=\frac{5.40-4.15}{0.5}$

$Z=\frac{1.25}{0.5}=2.5$

Thus, a net interest margin of 5.40 percent stands at 2.5 standard deviations above the average.

From the standard normal distribution table, the area to the left of 2.5 is 0.9938. Hence, the likelihood of a randomly selected U.S. bank achieving a net interest margin greater than 5.40 percent is 1-0.9938=0.0062

(b) The z-value corresponding to 4.40 percent is $Z=\frac{4.40-4.15}{0.5}=0.5$ The net interest margin of 4.40 percent is situated at 0.5 standard deviation above the average.

According to the normal distribution table, the area to the left of 0.5 is 0.6915

Thus, the probability of a randomly chosen U.S. bank having a net interest margin below 4.40 percent equals 0.6915

(c) The z-value indicating 95% is 1.65

Substituting 1.65 into the equation enables us to find X. $1.65=\frac{X-4.15}{0.5}$

$1.65\times 0.5=X-4.15$ $0.825=X-4.15$

$0.825+4.15=X$

$4.975=X$

For a bank that wishes for its net interest margin to fall below that of 95 percent of all U.S. banks, it should aim for a net interest margin of 4.975 percent.