Question

Find the probability, rounded to four decimal places, assuming that x follows a normal distribution with a mean (μ) of 102 and a standard deviation (σ) of 14 for P(x >= 120).

Answer 1

To find the probability that x is greater than or equal to 120 when x follows a normal distribution with a mean of 102 and a standard deviation of 14, calculate the Z-score and then look up the corresponding probability. The probability that x is greater than or equal to 120 is approximately 10.03%.

Step-by-step explanation:

The student is asking to calculate the probability that a normally distributed variable is greater than or equal to a certain value. Given that the mean (μ) is 102 and the standard deviation (σ) is 14, we need to find P(x ≥ 120). To find this, we can use the Z-score formula:

Z = (X - μ) / σ

For X = 120, Z = (120 - 102) / 14 ≈ 1.2857. We then look up this Z-score in a standard normal distribution table or use a calculator to find the probability to the left of this Z-score. Since we want P(x ≥ 120), we need to subtract this value from 1 to find the probability to the right:

P(x ≥ 120) = 1 - P(Z < 1.2857)

Assuming P(Z < 1.2857) ≈ 0.8997, the probability becomes:

P(x ≥ 120) = 1 - 0.8997 ≈ 0.1003

Therefore, the probability that x is greater than or equal to 120 is approximately 0.1003 or 10.03%, rounded to four decimal places.