Question

question 13 (4 points) suppose x is a normally distributed random variable with mean μ = 55 and standard deviation σ = 4. using the normal curve, estimate the probability that x < 47. 99.7% 0.15% 2.5% 97.5%

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 55\), \(\sigma=4\), and \(x = 47\).

$$ z=\frac{47 - 55}{4}=\frac{-8}{4}=-2 $$

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution:

• Approximately 68% of the data lies within \(z=\pm1\)
• Approximately 95% of the data lies within \(z=\pm2\)
• Approximately 99.7% of the data lies within \(z=\pm3\)

The area to the left of \(z=-2\) (since we want \(P(X < 47)\)):
The total area under the normal curve is 1. The area within \(z=-2\) and \(z = 2\) is 95%. So the area outside of \(z=-2\) and \(z = 2\) is \(1-0.95 = 0.05\). Since the normal distribution is symmetric, the area to the left of \(z=-2\) is \(\frac{1 - 0.95}{2}=0.025\) or 2.5%

Answer:

2.5%