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Math  /  Data & Statistics

QuestionAssume that xx has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) 〔 USE SALT μ=106;σ=20P(x≥90)=\begin{aligned} \mu & =106 ; \sigma=20 \\ P(x \geq 90) & = \end{aligned}

Studdy Solution

STEP 1

1. The random variable x x follows a normal distribution with mean μ=106 \mu = 106 and standard deviation σ=20 \sigma = 20 .
2. We need to find the probability P(x≥90) P(x \geq 90) .

STEP 2

1. Standardize the normal variable.
2. Use the standard normal distribution table to find the probability.

STEP 3

Calculate the z-score for x=90 x = 90 using the formula:
z=x−μσz = \frac{x - \mu}{\sigma}
Substitute the given values:
z=90−10620=−1620=−0.8z = \frac{90 - 106}{20} = \frac{-16}{20} = -0.8

STEP 4

Find the probability P(z≥−0.8) P(z \geq -0.8) using the standard normal distribution table. This is equivalent to finding 1−P(z<−0.8) 1 - P(z < -0.8) .
From the standard normal distribution table:
P(z<−0.8)≈0.2119P(z < -0.8) \approx 0.2119
Therefore:
P(z≥−0.8)=1−0.2119=0.7881P(z \geq -0.8) = 1 - 0.2119 = 0.7881
The probability P(x≥90) P(x \geq 90) is:
0.7881 \boxed{0.7881}

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