Download Important properties of sampling distributions of sample means

Sec. 5.4 – Sampling Distributions and the Central Limit Theorem
A ____________ _____________ is the probability distribution of a sample
statistic that is formed when samples of size 𝑛 are repeatedly taken from a
population.
If the sample statistic is the sample mean, then the distribution is the
_____________ __________ of ___________ _____________. Every sample
statistic has a sampling distribution.
Important properties of sampling distributions of sample means:
1. The mean of the sample means ______ is equal to the population mean _____.
𝜇𝑥̅ = 𝜇
2. The standard deviation of the sample means ______ is equal to the population
standard deviation ______ divided by the square root of _____. This is also called
the ____________ _____________ of the mean.
𝜎𝑥̅ =
𝜎
√𝑛
The Central Limit Theorem
1. If 𝑛 ≥ 30 then the sampling distribution of sample means is approximately
normally distributed no matter what distribution represents the actual
population.
2. For any size 𝑛, if the population is normally distributed, then the sampling
distribution is also normally distributed.
Any Population Distribution
Normal Population Distribution
Distribution of sample means if 𝑛 ≥ 30
Distribution of sample means (for any
size 𝑛)
Phone bills for residents of a city have a mean of $64 and a standard deviation of
$9. Random samples of 36 phone bills are drawn from this population and the
mean of each sample is determined. Find the mean and standard error of the
mean of the sampling distribution.
The heights of fully grown white oak trees are normally distributed, with a mean
of 90 feet and standard deviation of 3.5 feet. Random samples of size 4 are drawn
from this population, and the mean of each sample is determined. Find the mean
and standard error of the mean of the sampling distribution.
The mean room and board expense per year of four-year colleges is $6803. You
randomly select 9 four-year colleges. What is the probability that the mean room
and board is less than $7088? Assume that the room and board expenses are
normally distributed, with a standard deviation of $1125.
A bank auditor claims that credit card balances are normally distributed, with a
mean of $2870 and a standard deviation of $900.
(1) What is the probability that a randomly selected credit card holder has a credit
card balance less than $2500?
(2 ) You randomly select 25 credit card holders .What is the probability that their
mean credit card balance is less than $2500?
Compare the probabilities from (1) and (2) and interpret your answer in terms of
the auditor’s claim in a complete sentence or two.