Question

Find the indicated probability assuming that x is a random variable with a normal distribution with the given mean and standard deviation. (Round your answer to four decimal places.) P(x ≤ 175), = 140, = 35

Answer 1

To find the probability P(x ≤ 175) for a random variable x with a normal distribution, mean of 140, and standard deviation of 35, calculate the Z-score, then use a normal distribution table or calculator to find the corresponding cumulative probability, which is approximately 0.8413 after rounding.

Step-by-step explanation:

To calculate the probability P(x ≤ 175) for a random variable x with a normal distribution, a mean (μ) of 140, and a standard deviation (σ) of 35, you can use the Z-score formula:

Z = (X - μ) / σ

Given X = 175, μ = 140, and σ = 35, inserting the values into the formula gives:

Z = (175 - 140) / 35

Z = 1

Now, use a normal distribution table or a calculator with normal distribution functions to find the probability P(Z ≤ 1). The value found here represents P(x ≤ 175).

For example:

P(Z ≤ 1) = normalcdf(-∞,1,0,1)

The cumulative probability from negative infinity to a Z-score of 1 will give us our desired probability. This should be approximately 0.8413, and when rounded to four decimal places, it is 0.8413.