Sampling Distributions and The Central Limit Theorem
Notes
Recall that sample statistics vary from random sample to random
sample. We will now study the relationship between the population
mean and the various sample means.
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Sample means vary from one another and from the population
mean.
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Sampling distributions of sample means are formed when taking
random samples of size n from a population.
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The Central Limit Theorem describes the properties of the
sampling distribution of size n.
Sampling Distributions
Notes
A random sample of size n has various descriptive statistics. The
probability distribution of the sample means from a population have
the following properties:
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The average of all possible x̄’s is the mean of the sample means,
µx̄ . It is equal to the population mean.
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The standard deviation of the sample means is σx̄ is equal to the
population standard deviation divide by the square root of the
√
sample size n. This is called the standard error of the mean.
Sampling Distributions
Example: You have a six-sided die with sides numbered
{1, 2, 3, 4, 5, 6}. The value x is the value of the face after rolling, and
it has mean µ = 3.5 and standard deviation σ = 1.708. Then you
randomly roll the die twice.
Construct the possible sample means of size n = 2. How can you find
the probability of each of these events?
Notes
The Central Limit Theorem
If a sample of size n ≥ 30 is drawn from any population with a mean
µ and a standard deviation σ, then
the sampling distribution of the sample mean is approximately
normal. Large values of n give better approximations.
Notes
If the population is known to have a normal distribution, then
the sampling distribution of the sample mean is itself normal for any
sample size.
In any case,
µx̄ = µ
and
σ
σx̄ = √ .
n
Example
Notes
A particular high school claims that its students perform unusually
high on the critical reading portion of the SAT. Nationwide, this test
has a mean score of 501, with a standard deviation of 112. The high
school takes a random sample of 50 students that have taken the
test, and their mean score was 515. Is this unusual?
We use the Central Limit Theorem to answer this question. Note:
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n = 50 is large enough to use the Central Limit Theorem.
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The population does not need to assume a normal distribution.
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What is the mean and standard deviation of the sampling
distribution?
Example
You have data for the length of time people spend driving each day
(www.udot.gov). Drivers age 15-19 spend an average of 25 minutes
driving each day. Assume σ = 1.5 minutes. You randomly select 50
drivers in this age group.
What is the distribution of the sampling distribution for the sample
mean?
What is the probability that the mean time they spend driving each
day is between 24.7 and 25.5 minutes?
If the sample is not randomly drawn, but instead drawn by using a
cluster sample, would you expect the probabilities in the last question
to be the same? Why or why not?
Notes
Example
Notes
The Central Limit Theorem can also be used to investigate unusual
events. An unusual event is one that occurs with a probability of less
than 5%.
A machine filling half-gallon cartons on milk is regulated so the
amount of milk dispensed has a mean of 64 ounces and a standard
deviation of 0.11 ounce. You randomly inspect 40 cartons and
measure the contents. The sample mean of the cartons of milk is
64.05 ounces.
Does this sample lie in an interval of unusual events?
What is the lowest mean so that sample means are in the top 5%?
Assignment
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Notes
§5.4: 6, 8, 21, 26, 33, 37
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Read §5.4
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Chapter Five Quiz, p 297 # 9, #10
Suggested Exercises:
Notes