Question

Let X be normally distributed with mean μ = 3,800 and standard deviation σ = 2,000. [You may find it useful to reference the z table.]

a. Find x such that P(X ≤ x) = 0.9382. (Round "z" value to 2 decimal places, and final answer to nearest whole number.)

b. Find x such that P(X > x) = 0.025. (Round "z" value to 2 decimal places, and final answer to nearest whole number.)

c. Find x such that P(3,800 ≤ X ≤ x) = 0.1217. (Round "z" value to 2 decimal places, and final answer to nearest whole number.)

d. Find x such that P(X ≤ x) = 0.4840. (Round "z" value to 2 decimal places, and final answer to nearest whole number.)

Answer 1

Solution :

a)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.9382      
          
Then, using table or technology,          
          
z =    1.54      
          
As x = u + z * s,          
          
where          
          
u = mean = 3800      
z = the critical z score =    1.54      
s = standard deviation = 2000      
          
Then          
          
x = critical value = 6880 [ANSWER]

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b)

First, we get the z score from the given left tailed area. As          
          
Left tailed area = 1 - 0.025 =   0.975      
          
Then, using table or technology,          
          
z =    1.96      
          
As x = u + z * s,          
          
where          
          
u = mean = 3800      
z = the critical z score =    1.96      
s = standard deviation = 2000      
          
Then          
          
x = critical value = 7720 [ANSWER]

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c)

First, we get the z score from the given left tailed area. As the lower endpoint, 3800, is the mean, then it has a left tailed area of 0.5. Thus, the left tailed area of x is          
          
Left tailed area = 0.50 + 0.1217 =   0.6217      
          
Then, using table or technology,          
          
z =    0.31      
          
As x = u + z * s,          
          
where          
          
u = mean = 3800      
z = the critical z score =    0.31      
s = standard deviation = 2000      
          
Then          
          
x = critical value = 4420 [ANSWER]

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d)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.484      
          
Then, using table or technology,          
          
z =    -0.04      
          
As x = u + z * s,          
          
where          
          
u = mean = 3800      
z = the critical z score =    -0.04      
s = standard deviation = 2000      
          
Then          
          
x = critical value = 3720 [ANSWER]  

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