AP Statistics
Sampling Distribution of Mean
Name:___________________________________
1. Suppose that we randomly select a sample of 64 measurements from a population having a mean equal to 20
and a standard deviation equal to 4.
a. Find the mean and standard deviation of the sampling distribution of the sample mean.
𝜇𝑥̅ = 20
σ=
4
√64
1
= 2 = .5
b. Calculate the probability that we will obtain a sample mean greater than 21.
𝜇𝑥̅ = 20
σ = .5
𝑥̅ = 21
Normcdf(21, 1E99, 20, 0.5) = 0.0228
2. The average length of a hospital stay in the US is 9 days with a standard deviation of 3 days. Assume a simple
random sample of 100 patients is obtained and the mean stay is them is calculated.
a. What is the mean of the sampling distribution for the sample of 100 patients?
𝜇𝑥̅ = 9
b. What is the standard deviation of the sampling distribution of the mean for the sample of size 100?
σ=
3
√100
=
3
10
= .3
c. What is the probability of obtaining a sample mean less than 9.6 days?
𝜇𝑥̅ = 20
σ = .5
𝑃 (𝑥 < 9.6)
Normcdf(-1E99, 9.6, 9, .3) = 0.977
3. The scores of students on the ACT college entrance exam in a recent year had a normal distribution with a mean
of 18.6 and a standard deviation of 5.9.
a. What is the probability that a single student randomly chosen from all those taking the test scores 21 or
higher on their test?
n = 1,
𝜇𝑥̅ = 18.6
σ=
5.9
=
√1
5.9
𝑃 (𝑥̅ > 21) = ?
Normcdf(21, 1E99, 18.6, 5.9) = .342
b. Now take a simple random sample of 50 students who took the test. What is the mean and standard
deviation of the sample mean of these 50 students?
n = 50, 𝜇𝑥̅ = 18.6
σ=
5.9
√50
= . 834
c. What is the probability that the mean score of these students is 21 or higher?
n = 1,
𝜇𝑥̅ = 18.6
σ=
5.9
√50
= . 834
𝑃 (𝑥̅ > 21) = ?
Normcdf(21, 1E99, 18.6, .834) = .00200
4. Weekly postage expenses for your company have a mean of $325 and a standard deviation of $54. You
company is allowed for $350 postage per week in its budget.
a. What is the probability that the average weekly postage expense for the past 9 months (assuming each
month is 4 weeks) exceed the budgeted amount of $350?
𝜇𝑥̅ = $325
σ=
54
√36
=9
𝑃 (𝑥̅ > 350) = ?
Normalcdf(350, 1E99, 325, 9) = 0.00274
b. The recent problems with the stock market are affecting your business and you need to reallocate some
of your funds, so you’d like to reduce your postage expenses. What is the probability that your postage
for the next week will not exceed $330?
Cannot be calculated since we do not know if the population of weekly postage expenses is normally
distributed.