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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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& www.apsu.edu/jonesmatt/1530.htm

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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. & www.apsu.edu/jonesmatt/1530.htm 2

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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.

& www.apsu.edu/jonesmatt/1530.htm

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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! & www.apsu.edu/jonesmatt/1530.htm

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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.

& www.apsu.edu/jonesmatt/1530.htm

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Sample Proportions

It is reasonable to estimate the population proportion p with the sample proportion

pˆ ≡

# of successes x = n sample size n

However, before drawing a sample from a population, the sample proportion pˆ is a random variable, so it has a distribution. Knowing the distribution of the sample proportion pˆ helps us to know ‘how well’ pˆ can be used to approximate p.

& www.apsu.edu/jonesmatt/1530.htm

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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 pˆ = 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 pˆ = = 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. & www.apsu.edu/jonesmatt/1530.htm

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' Sample Proportion Statistics Because proportions are means, the following results flow from the results on slide #5: Since pˆ is a kind of x and p is a kind of µ, µpˆ = p Since

p

p(1 − p) is the std. dev. for one trial for a binomial experiment, σ σpˆ = √ = n

p

p(1 − p) √ = n

r

p(1 − p) n

Here, as with the case for means, σpˆ is often called the standard error. Note the larger the sample size n, the smaller the standard error is when estimating p by pˆ. & www.apsu.edu/jonesmatt/1530.htm

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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.

& www.apsu.edu/jonesmatt/1530.htm

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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.

& www.apsu.edu/jonesmatt/1530.htm

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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%?

& www.apsu.edu/jonesmatt/1530.htm

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