Final answer:
To find P(x ≥ 13.5) for a random variable with a mean of 10 and a standard deviation of 2, calculate the z-score to be 1.75, use a standard normal distribution table or calculator to find P(z < 1.75) ≈ 0.9599, and then calculate 1 - 0.9599 to get P(x ≥ 13.5) as approximately 0.0401 or 4.01%.
Step-by-step explanation:
To find P(x ≥ 13.5) for a random variable with a mean (μ) of 10 and a standard deviation (σ) of 2, we first need to calculate the z-score for x = 13.5. The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the z-score is:
z = (x - μ) / σ
Plugging in our values we get:
z = (13.5 - 10) / 2
z = 1.75
Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability that x is greater than or equal to 13.5. The standard normal distribution is symmetric about zero, so we look up the probability for z = 1.75 and subtract it from 1 (since the table gives us the probability that z is less than a particular value).
Using a standard normal distribution table or a calculator:
P(z < 1.75) ≈ 0.9599
Therefore, P(x ≥ 13.5) = 1 - P(z < 1.75)
P(x ≥ 13.5) ≈ 1 - 0.9599
P(x ≥ 13.5) ≈ 0.0401
Thus, the probability that x is 13.5 or more is approximately 0.0401 or 4.01%.