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Find the value of x so that the area under the normal curve is between...

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3 Answers

Hi An,
You didn't provide us with a value of μ or σ, so I'll just leave it as μ and σ in the answer:
1. Since x is less than μ we have a negative z score. The z score which corresponds to a probability of 0.4525 between x and μ is -1.67 (you can find this using a table, or using the excel function "=NORMINV(0.5 - 0.4525, 0, 1)", which computes the z score for an area of 0.4525 between x and μ).
Therefore the desired value of x is:
x = μ - 1.67σ
or if μ = 0 and σ = 1,
x = -1.67
2. I'll leave this one up to you, but its the same as (1), except that you're adding the area to 0.5 in the inverse normal function instead of subtracting it, which will give you a positive z score.


I know this isn't worth much but in my book, the answer for number 1 is 158.25.
Also, the equation is:
x = μ + zσ
Hi An,
Since you didn't provide a value for μ or σ we can't really come up with a value for x, only a z score corresponding to the area you provided. The values for μ and σ should have been provided by your book for the question (or they at least should have given you the information necessary to compute μ and σ).
Remember that P(Z≤µ) = P(Z≥µ) = 0.5.
a) P(Z≤z) = 0.5-0.4525 = 0.0475 ⇒ z=-1.67
b) P(Z≥z) = 0.5+0.48 = 0.98 ⇒ z=2.05
To find x, we need to know the values of µ and σ, since x =μ + zσ.
1. To the left of μ, the area is 0.5, and of that, 0.4525 is between x and μ.
    Hence, the area  left of x is 0.5 - 0.4525 = 0.0475.
    Now you can look up the table of z-scores to get z = -1.67.
    Hence x = μ - 1.67σ.
2. Since 0.4800 is between μ and x, the area left of x is 0.5+0.4800 = 0.9800.
    Now you can look up the table of z-scores to get z = 2.054.
    Hence x = μ + 2.054σ.