Question

A simple random sample of size n=81 is obtained from a population that is skewed right with mean =80 and std dev=18. What is p(x less than equal to 75)?

Answer 1

To find the probability of X being less than or equal to 75 in a simple random sample of size 81, we need to use the properties of the normal distribution.

Given:

Sample size (n) = 81

Population mean (μ) = 80

Population standard deviation (σ) = 18

First, we need to calculate the standard error of the mean (SE), which is given by:

SE = σ / sqrt(n)

SE = 18 / sqrt(81)

SE = 18 / 9

SE = 2

Next, we standardize the value of 75 using the formula:

Z = (X - μ) / SE

Z = (75 - 80) / 2

Z = -5 / 2

Z = -2.5

Now, we can find the probability of Z being less than or equal to -2.5 using a standard normal distribution table or a calculator. From the table or calculator, we find that the probability is approximately 0.0062.

Therefore, the probability of X being less than or equal to 75 in the given simple random sample is approximately 0.0062 or 0.62%.

Explanation: