µ = 4.9; s = 1.6 P(3 = x = 6) =

Question

 

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)

Answer 1

Given: μ = 4.9, σ = 1.6

Find: P(3≤ x ≤6)

 

Method:

Since we have a normal distribution with specified mean and standard deviation, we can determine the probability using Z-scores. So we will need to reference the table for Z-scores. From there, we can look up the probabilities for x=3 and x=6. Subtract the two probabilities and then we can find our final answer.

 

1. Convert the x-values into Z-scores.

 

Z = (x - μ)/σ

 

Z(x=3) = (3 - 4.9)/1.6 = -1.1875

Z(x=6) = (6 - 4.9)/1.6 = +0.6875

 

2. Use the Z-score statistical table to determine the probabilities at each calculated Z-score.

 

P(Z = -1.1875) = 0.1175

P(Z = +0.6875) = 0.7541

 

Note: The Z-score table only goes to two decimal places for the Z-score. You can interpolate the specific probabilities using a linear approximation. See the example after the answer.

 

3. Subtract the two probabilities.

P(3 ≤ x ≤ 6)
= P(-1.1875 ≤ Z ≤ 0.7451)
= P(Z ≤ 0.7451) - P(-1.1875 ≤ Z)
= 0.7541

4. Answer: P(3 ≤ x ≤ 6) = 0.7541

 

 

 

* * * * * * * * * * * * *

How to calculate the probabilities for Z-scores in the range of values for the table but not specifically given:

 

Z = -1.1875 lies between the two values in the table for Z=-1.18 and Z=-1.19

 

[P(Z≤-1.1875) - P(Z≤-1.18)] / [P(Z≤-1.19) - P(Z≤-1.18)] = [-1.1875 - (-1.18)] / [-1.19 - (-1.18)]

 

From the table

• P(Z ≤ -1.18) = 0.1190
• P(Z ≤ -1.19) = 0.1170

 

Solve for P(Z=-1.1875). Repeat for P(Z=+0.6875)