Question:
The average life expectancy of people in the United States in 2018 was 78.54 years with a standard deviation of approximately 15 years.
Suppose we take a sample of 40 individuals in the United States, what are the mean and standard deviation for sample averages from samples of size 40?
Answer:
[tex]\bar x = 78.54[/tex]
[tex]\sigma_x = 2.37[/tex]
Step-by-step explanation:
Given
[tex]\mu = 78.54[/tex] --- Population mean
[tex]\sigma = 15[/tex] --- Population standard deviation
[tex]n = 40[/tex] --- Sample size
Solving (a): The sample mean
According to the central limit theorem, sample mean is always equal to the population mean.
So:
[tex]\bar x = 78.54[/tex]
Solving (b): The sample standard deviation
Using the central limit theorem, we have:
[tex]\sigma_x = \frac{\sigma}{\sqrt n}[/tex]
So:
[tex]\sigma_x = \frac{15}{\sqrt{40}}[/tex]
[tex]\sigma_x = \frac{15}{6.32}[/tex]
[tex]\sigma_x = 2.37[/tex]
