A sample of 50 retirees is drawn at random from a normal population whose mean a
ID: 3132354 • Letter: A
Question
A sample of 50 retirees is drawn at random from a normal population whose mean age and standard deviation are 75 and 6 years, respectively.
Describe the shape of the sampling distribution of the sample mean in this case?
Find the mean and standard error of the sampling distribution of the sample mean.
What is the probability that the mean age exceeds 73 years?
What is the probability that the mean age is at most 73 years?
What is the probability that two randomly selected retirees are over 73 years of age?
Explanation / Answer
Describe the shape of the sampling distribution of the sample mean in this case?
It is normally distributed (bell shaped) by the central limit theorem. *****************************************
Find the mean and standard error of the sampling distribution of the sample mean.
It is with the same mean as population mean,
u(X) = u = 75
and a reduced standard deviation,
sigma(X) = sigma/sqrt(n) = 6/sqrt(50) = 0.848528137.
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What is the probability that the mean age exceeds 73 years?
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 73
u = mean = 75
n = sample size = 50
s = standard deviation = 6
Thus,
z = (x - u) * sqrt(n) / s = -2.357022604
Thus, using a table/technology, the right tailed area of this is
P(z > -2.357022604 ) = 0.990788937 [ANSWER]
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What is the probability that the mean age is at most 73 years?
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 73
u = mean = 75
n = sample size = 50
s = standard deviation = 6
Thus,
z = (x - u) * sqrt(n) / s = -2.357022604
Thus, using a table/technology, the left tailed area of this is
P(z < -2.357022604 ) = 0.009211063 [ANSWER]
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What is the probability that two randomly selected retirees are over 73 years of age?
FOR ONE RETIREE:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 73
u = mean = 75
s = standard deviation = 6
Thus,
z = (x - u) / s = -0.333333333
Thus, using a table/technology, the right tailed area of this is
P(z > -0.333333333 ) = 0.63055866
Hence, for two retirees,
P(both over 73) = 0.63055866^2 = 0.397604224 [ANSWER]