X has a normal distribution with the given mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) 𝜇 = 57, 𝜎 = 10, find P(32 ≤ X ≤ 62)

Question

You may need to use the appropriate appendix table or technology to answer this question.

X has a normal distribution with the given mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)

𝜇 = 57, 𝜎 = 10, find P(32 ≤ X ≤ 62)

Answer 1

Lets start by writing down all our information.

μ = 57

σ = 10

μ and σ have a normal distribution.

32≤X≤62

The first thing we need to do is calculate a Z score for X = 32 and X = 62. The formula is Z = (x-μ)/σ.

Plugging in 32 and 62 for X we get Z₁ = -2.5 and Z₂ = 0.5. Now the Z score is a measure of how many standard deviations x is from a Normalized normal distribution (μ = 0, σ = 1). The probability is the area under this normalized normal distribution. Using a Z table or a calculator (or google) we can look up the area under the curves for Z≤-2.5 and Z≤0.5. We get 0.0062 for Z₁ and 0.6915 for Z₂. Now those values are the areas under the curve for Z≤-2.5 (X≤-32) and Z≤0.5 (X≤62). We want the area in between so that is just 0.6915 - 0.0062 = 0.6853 or 68.53%.