Question

A consensus forecast is the average of a large number of individual analysts' forecasts. Suppose the individual forecasts for a particular interest rate are normally distributed with a mean of 6 percent and a standard deviation of 1.6 percent. A single analyst is randomly selected. Find the probability that his/her forecast is

Round your answers to 4 decimal places.

(a) At least 3.4 percent.

(b) At most 8 percent.

(c) Between 3.4 percent and 8 percent.

Answer 1

Solution :

Given that ,

mean = = 6% = 0.06

standard deviation = = 1.6% = 0.016

(a) 3.4% = 0.034

P(x 0.034) = 1 - P(x   0.034)

= 1 - P((x - ) / (0.034 - 0.06) / 0.016)

= 1 -  P(z -1.625)  

= 1 - 0.0521

= 0.9479

P(x 0.034) = 0.9479

Probability = 0.9479

(b) 8% = 0.08

P(x 0.08) = P((x - ) / (0.08 - 0.06) / 0.016)

= P(z 1.25)

Using standard normal table,

P(x 0.08) = 0.8944

Probability = 0.8944

(c)

P(0.034 < x < 0.08) = P((0.034 - 0.06)/ 0.016) < (x - ) / < (0.08 - 0.06) / 0.016) )

= P(-1.625 < z < 1.25)

= P(z < 1.25) - P(z < -1.625)

= 0.8944 - 0.0521 = 0.8423

Probability = 0.8423