Answer:
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: