Question

When returns from a project can be assumed to be normally distributed, such as those shown in Figure 13-6 (represented by a symmetrical, bell-shaped curve), the areas under the curve can be determined from statistical tables based on standard deviations. For example, 68.26 percent of the distribution will fall within one standard deviation of the expected value ( D¯¯¯D¯ ± 1σ). Similarly, 95.44 percent will fall within two standard deviations ( D¯¯¯D¯  ± 2σ), and so on. An abbreviated table of areas under the normal curve is shown next.
  

Number of σ's From Expected Value			+ or –		+ and –
	0.50		0.1915		0.3830
	1.00		0.3413		0.6826
	1.50		0.4332		0.8664
	2.00		0.4772		0.9544
	3.46		0.4997		0.9994

Assume Project A has an expected value of $20,000 and a standard deviation (σ) of $4,000.

a. What is the probability that the outcome will be between $18,000 and $22,000? (Round your answer to 4 decimal places.)
  

  
b. What is the probability that the outcome will be between $16,000 and $24,000? (Round your answer to 4 decimal places.)
  

  
c. What is the probability that the outcome will be at least $12,000? (Round your answer to 4 decimal places.)
  

  
d. What is the probability that the outcome will be less than $33,830? (Round your answer to 4 decimal places.)
  

  
e. What is the probability that the outcome will be less than $18,000 or greater than $22,000? (Round your answer to 4 decimal places.)
  

Answer 1

0.50		0.1915		0.3830
	1.00		0.3413		0.6826
	1.50		0.4332		0.8664
	2.00		0.4772		0.9544
	3.46		0.4997		0.9994

Let x be the returns from a project .

Given : mean ( µ ) = $20,000 and standard deviation (σ) = $4,000

a. What is the probability that the outcome will be between $18,000 and $22,000? (Round your answer to 4 decimal places.)
  

P( $18,000 ≤x ≤ $22,000) =

=

= P( -0.5 ≤ z  ≤ 0.5 )

= 0.3830 --- ( from the abbreviated table of areas under the normal curve )

b. What is the probability that the outcome will be between $16,000 and $24,000? (Round your answer to 4 decimal places.)

P( $16,000 ≤x ≤ $24,000) =

=

= P( -1 ≤ z  ≤ 1 )

= 0.6826 --- ( from the abbreviated table of areas under the normal curve )

  

c. What is the probability that the outcome will be at least $12,000? (Round your answer to 4 decimal places.)
  

P( x ≥ $12000 )

=

=P( z ≥ -2 )

= 0.4772 + 0.5000

= 0.9772
  
d. What is the probability that the outcome will be less than $33,830? (Round your answer to 4 decimal places.)
  

P( x < $33830 )

=  

= P( z  ≤ 3.46 )

= 0.5000 + 0.4997

= 0.9997

e. What is the probability that the outcome will be less than $18,000 or greater than $22,000? (Round your answer to 4 decimal places.)

P( x < $18000 ) + P( x > $22000)

=1 - P( $18,000 ≤x ≤ $22,000)

= 1 - 0.3830

= 0.6170