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 1.832 in currency A​ (to currency​ B) and standard deviation 0.044 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). ​a) What would the cutoff rate be that would separate the highest 2.5​% of currency​ A/currency B​ rates?

Answer 1

The cutoff rate that would separate the highest 2.5​% of currency​ A/currency B​ rates is 1.92.

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 1.832

Standard deviation = 0.044

Top 2.5%

95% of the measures are within 2 standard deviation of the mean.

Since the normal distribution is symmetric, this 95% goes from the 50 - 95/2 = 2.5th percentile to the 50 + 95/2 = 97.5th percentile.

The 97.5th percentile is the cutoff for the highest 2.5% of currency​ A/currency B​ rates, and it is 2 standard deviations above the mean.

1.832 + 2*0.044 = 1.92

The cutoff rate that would separate the highest 2.5​% of currency​ A/currency B​ rates is 1.92.