Question

The proportion of values of X between -3 and 8, if ∼(5,4.2)
X
∼
N
(
5
,
4.2
)
. Round your answer to three decimal places.

Answer 1

If I'm understanding the question correctly, you're saying that X is a random variable following a normal distribution with mean 5 and standard deviation 4.2, and you want to find the proportion of the distribution that lies in the interval (-3, 8). In other words, the probability

P(-3 < X < 8)

Transform X to Z, which follows the standard normal distribution with mean 0 and standard deviation 1, using the rule

X = µ + σZ   ==>   Z = (X - µ)/σ

where µ and σ are the mean and standard devation of X, respectively.

We have

P(-3 < X < 8) = P((-3 - 5)/4.2 < (X - 5)/4.2 < (8 - 5)/4.2)

… ≈ P(-1.904 < Z < 0.714)

… ≈ P(Z < 0.714) - P(Z < -1.904)

… ≈ 0.762 - 0.028 ≈ 0.734

Answer 2

The proportion of values of X between - 3 and 5 is 0.734

The expression ∼(5,4.2) gives the mean and standard deviation of X

Mean, μ = 5

Mean, μ = 5Standard deviation, σ = 4.2

Using the normal distribution :

The proportion of values of X is given as :

z = (x - μ) / σ

P(Z < z) - P(Z < z)

P(Z < (x - μ) / σ) - P(Z < (x - μ) / σ)

For X = - 3 and x = 8

P(Z < (8 - 5) / 4.2) - P(Z < (-3 - 5) / 4.2)

P(Z < 0.7143) - P(Z < - 1.90476)

0.76248 - 0.0284 = 0.73408

Hence, the proportion of values of X between - 3 and 5 is 0.734

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