Download The Distributions of the Sample Mean and Sample Proportion

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The Distributions of the Sample Mean and
Sample Proportion
1 Sample Means
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2 Sample Proportions
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3 Central Limit Theorem
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Sample Means
It is reasonable to estimate the population mean µ with the sample mean x:
However, before drawing a sample from a population, the sample mean X is
a random variable, so it has a distribution.
Knowing the distribution of the sample mean X helps us to know ‘how well’
X can be used to approximate µ.
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Example 1 The heights in inches of five starting players on a men’s
basketball team are
Alfred: 76
Bob: 79
Carl: 85
Dennis: 82
Edgar: 78
How many samples of two players can be taken?
List all possible samples and the sample means.
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The mean of the sample means is
The mean height is
This is not a coincidence! The mean of all the sample means is always the
mean of the population.
The population standard deviation of the players’ heights is
The standard deviation of the mean heights is
These numbers are different!
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Sample Mean Statistics
The mean of the sample means is equal to the population mean:
µx = µ
If sampling from an infinite population, the standard deviation of the sample
mean (which is often called the standard error) is
σ
σx = √
n
The above can be shown with A LOT of algebra... we’ll skip the details.
The larger the sample size n, the smaller the standard error tends to be in
estimating µ by x.
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Sample Proportions
It is reasonable to estimate the population proportion p with the sample
proportion
p̂ ≡
# of successes
x
=
n
sample size n
However, before drawing a sample from a population, the sample proportion
p̂ is a random variable, so it has a distribution.
Knowing the distribution of the sample proportion p̂ helps us to know ‘how
well’ p̂ can be used to approximate p.
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Sample Proportions
GUESS WHAT? Proportions are means.
Example 2 In a survey, 55 of 150 fans favored Sammy Hagar over David
Lee Roth. Of course the sample proportion is
55
p̂ =
150
Another way to see this is to assign a ‘1’ to each vote for Sammy, and a ‘0’
to each vote for Roth. Then
55 1s
95 0s
z
}|
{ z
}|
{
1 + 1 + ··· + 1+0 + 0 + ··· + 0
55
p̂ =
=
150
150
Then a proportion is a mean of 1s and 0s, and so the results for the
distribution of the sample mean also apply to proportions.
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Sample Proportion Statistics
Because proportions are means, the following results flow from the results on
slide #5:
Since p̂ is a kind of x and p is a kind of µ,
µp̂ = p
Since
p
p(1 − p) is the std. dev. for one trial for a binomial experiment,
σ
σp̂ = √ =
n
p
p(1 − p)
√
=
n
r
p(1 − p)
n
Here, as with the case for means, σp̂ is often called the standard error.
Note the larger the sample size n, the smaller the standard error is when
estimating p by p̂.
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Central Limit Theorem
This is one of the most famous and important results in all of mathematics.
It forms the basis for many statistical tests.
Theorem 3 As the sample size n approaches ∞, the random variable X
becomes distributed more and more like a normal with mean µ and standard
√
deviation σ/ n.
This result is true for every population distribution with finite standard
deviation.
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So What Does the Central Limit Theorem Imply?
The ages of pennies tend to have a right-skewed distribution. We’d like to
estimate the mean age of a penny in circulation. Using Minitab, compare the
distributions of sample means for samples of size 5 and size 40.
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Examples
Example 4 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 25 cups, what’s the approximate
probability the sample mean amount of pudding is less than 100.5g? What’s
approximate probability it will be between 100.6g and 101.4g?
Example 5 About 72% of all Halloween candy is undesirable. If you take
50 pieces at random, what is the probability that the undesirable proportion
in your sample is between 70% and 74%? What’s the chance the undesirable
proportion of candy in a sample of size 80 is between 70% and 74%?
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