Download Chapter 7: Sampling Distributions Distributions of the Sample Mean

Chapter 7: Sampling Distributions
Distributions of the Sample Mean and the Sample Proportion
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Distribution of the Sample Mean:
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We usually estimate the population mean µ by the sample mean x .
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Once we continue sampling, the sample mean becomes a random variable (takes different
values for each sample)
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It is important to know the distribution of the random variable
properties.
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Knowing the distribution of the sample mean X helps us to know `how well'
used to approximate the population mean µ (the true mean).
X
in order to investigate its
X
can be
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Example: The heights of a basketball team of five starting players are
Alfred: 76
Bob: 79
Carl: 85
Dennis: 82
Edgar: 78
a) Calculate the mean µ and standard deviation σ of the heights.
b) List all possible samples of size 4 and then compute the sample mean for each sample.
c) Calculate the mean of all sample means in part b.
d) What do you notice?
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Remarks:
Mean(X ) = µ
- The mean of all the sample means is always the mean of the population. i.e.
- The standard deviation of all the sample means is always the standard deviation of the
population divided by the sample size. i.e.
Standard deviation(X ) = σ
n
- From the previous equation, the larger the sample size n, the smaller the standard error (or
the standard deviation). Which means that, the larger the sample size, the better the sample mean
x
in estimating the population mean
µ.
Example: The second exam scores for 1530 class is normally distributed with mean of 65 and
standard error of 11 . A sample size of 30 was taken from the class. What is the mean and the
standard deviation for the random variable X of size 30?
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Distribution of the Sample Proportion:
Examples of proportions: The proportion of males in USA is 0.54 (or 54%). The proportion of
Cancer in a country is 0.002 (or 0.2%).
We denote to the population proportion by p. One we sample from the population, we can
estimate the population proportion p by the sample proportion p̂ .
Example: A researcher attempts to estimate the proportion of males in USA, he took a sample
with size 1000 residents in USA and found out that 562 were males. What is the sample
proportion of males in this example?
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The sample proportion
p̂ =
frequency of catogery
sample size
Once we continue sampling, the sample proportion becomes a random variable (takes
different values for each sample)
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-
It is important to know the distribution of the random variable
p̂ in order to investigate its
properties.
-
Knowing the distribution of the sample proportion
used to approximate the population proportion
p
p̂ helps us to know `how well' p̂ can be
(the true proportion).
Remarks:
- The mean of all the sample proportions is always equals to the population proportion
p . i.e.
Mean(pˆ ) = p
- The standard error of all the sample proportions is always equals to the standard error of the
population proportion divided by the square root of the sample size. i.e.
Standard error( pˆ ) = p(1 − p)
n
- From the previous equations, the larger the sample size n, the smaller the standard error (or
the standard deviation). Which means that, the larger the sample size, the better the sample
proportion
p̂
in estimating the population proportion
p.
Example: The proportion of males in USA is p = 0.54 . A researcher took a sample with size
1000 residents in USA and found out that 562 were males. What is the mean and standard error
of the random variable p̂ ?
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Central Limit Theorem:
Central limit theorem is one of the most famous theorems in statistics. It is used in so many
applications in real life.
Central Limit Theorem: As the sample size n becomes large, the random variable
X becomes approximately normal distributed with mean µ and standard
deviation
σ .
n
Remark: Conventionally, sample size is large if it is greater than or equal 30. i.e. n ≥ 30.
Activity: Understanding central limit theorem.
Using Minitab, generate a 200 samples with size 40 from a normal population with µ=2 and
σ = 1 . Calculate the sample means and then graph a histogram for the sample means. What do
you notice?
Examples:
1) The package states a pudding cup contains 99g of pudding. Actually, the amount of pudding
in a cup has mean 101g and standard deviation 1g. If you take a sample of 35 cups,
a) What is the distribution of the sample mean amount of pudding? Give specific name.
b) What is the mean and standard deviation for X .
c) What theorem your answer in part (a) was based on?
d) What is the approximate probability that the sample mean amount of pudding is between
100.6g and 101.4 g
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2) About 72% of all Halloween candy is undesirable. If you take 50 pieces at random.
a) What is the distribution of the sample proportion p̂ ? Give specific name.
b) What is the mean and standard deviation for P̂ .
c) What is the probability that the undesirable proportion in your sample is between 70% and
74%?
d) What's the chance the undesirable proportion of candy in your sample is bigger than 65%?
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