Question

The daily exchange rates for the​ five-year period 2003 to 2008 between currency A and currency B are well modeled by a normal distribution with mean 1.814 in currency A​ (to currency​ B) and standard deviation 0.035 in currency A. Given this​ model, and using the​ 68-95-99.7 rule to approximate the probabilities rather than using technology to find the values more​ precisely, complete parts​ (a) through​ (d).

Question: ​a) What would the cutoff rate be that would separate the highest 2.5​% of currency​ A/currency B​ rates? The cutoff rate would be ___________ (type an integer or a decimal rounded to the nearest thousandth as needed)

Question: What would the cutoff rate be that would separate the highest 50% ? The cutoff rate would be _______________

Question: What would the cutoff rate be that would separate the middle 68% ? The lower cutoff rate would be ____________

Question: The upper cutoff rate would be ? ____________________

Question: What would the cutoff rate be that would separate the highest 16%? ________________

Answer 1

Using the​ 68-95-99.7 rule, we can write

68% values lie between (mean - standard deviation, mean + standard deviation) = (1.814 - 0.035, 1.814 + 0.035) = (1.779, 1.849)

95% values lie between (mean - 2*standard deviation, mean + 2*standard deviation) = (1.814 - 2*0.035, 1.814 + 2*0.035) = (1.744, 1.884)

99.7% values lie between (mean - 3*standard deviation, mean + 3*standard deviation) = (1.814 - 3*0.035, 1.814 + 3*0.035) = (1.709, 1.919)

a) (100-95) = 5% values lie outside the interval (1.744, 1.884)

Among them, 2.5% values lie above 1.884

The cutoff rate be that would separate the highest 2.5​% of currency​ A/currency B​ rates = 1.884

b) 50% values lie below the mean = 1.814 and 50% values lie above the mean = 1.814

The  cutoff rate be that would separate the highest 50% = 1.814

c) The cutoff rate be that would separate the middle 68% = (1.779, 1.849)

d) (100 - 68) = 32% values lie outside the interval (1.779, 1.849)

Among them, 16% values lie above 1.849

The  cutoff rate be that would separate the highest 16% = 1.849