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**Sampling Distributions**

A review by Hieu Nguyen (03/27/06)

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**Parameter vs Statistic**

A parameter is a description for the entire population. Example: A parameter for the US population is the proportion of all people who support President Bush’s nomination of Samuel Alito to the Supreme Court. p=.74

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**Parameter vs Statistic**

A statistic is a description of a sample taken from the population. It is only an estimate of the population parameter. Example: In a poll of 1001 Americans, 73% of those surveyed supported Alito’s nomination. p-hat=.73

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Bias The bias of a statistic is a measure of its difference from the population parameter. A statistic is unbiased if it exactly equals the population parameter. Example: The poll would have been unbiased if 74% of those surveyed approved of Alito’s nomination. p-hat=.74=p

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Sampling Variability Samples naturally have varying results. The mean or sample proportion of one sample may be different from that of another. In the poll mentioned before p-hat=.73. A repetition of the same poll may have p-hat=.75.

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**Central Limit Theorem (CLT)**

Populations that are wildly skewed may cause samples to vary a great deal. However, the CLT states that these samples tend to have a sample proportion (or mean) that is close to the population parameter. The CLT is very similar to the law of large numbers.

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CLT Example Imagine that many polls of 1001 Americans are done to find the proportion of those who supported Alito’s nomination. Although the poll results vary, more samples have a mean that is close to the population parameter μ=.74.

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CLT Example Plot the mean of all samples to see the effects of the CLT. Notice how there are more sample means near the population parameter μ=.74. This histogram is actually a sampling distribution

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**Sampling Distributions: Definition**

Textbook definition: A sampling distribution is the distribution of values taken by the statistic in all possible samples of the same size from the same population. In other words, a sampling distribution is a histogram of the statistics from samples of the same size of a population.

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**Two Most Common Types of Sampling Distributions**

Sample Proportion Distribution Distribution of the sample proportions of samples from a population Sample Mean Distribution Distribution of the sample means of samples from a population For both types, the ideal shape is a normal distribution

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**Sampling Distributions: Conditions**

Before assuming that a sampling distribution is normal, check the following conditions: Plausible Independence Randomness Each sample is less than 10% of the population

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**Sampling Distributions As Normal Distributions**

When all conditions met, the sampling distribution can be considered a normal distribution with a center and a spread. Note: With sample proportion distributions, another condition must be meet: Success-failure conditon – there must be at least 10 success and 10 failures according to the population parameter and sample size

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**Sampling Distributions As Normal Distributions: Equations**

Sample Proportion Distribution p = population proportion (given) Sample Mean Distribution μ = population mean (given) σ = population standard deviation (given)

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**Sampling Distributions As Normal Distributions: Note**

Note: If any of the parameters are unknown, use the statistics from a sample to approximate it.

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**Using Sampling Distributions**

Sampling Distributions can estimate the probability of getting a certain statistic in a random sample. Use z-scores or the NormalCDF function in the TI-83/84.

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**Using Sampling Distributions: Z-Scores w/ Example**

Use the z-score table to find appropriate probabilities Example: Find the probability that a poll of Americans that support Alito’s nomination will return a sample proportion of .72.

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**Using Sampling Distributions: NormalCDF Function w/ Example**

The syntax for the NormalCDF function is: NormalCDF(lower limit, upper limit, μ, σ) Example: Find the probability that a sample of size 25 will have a mean of 5 given that the population has a mean of 7 and a standard deviation of 3.

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**Sampling Distribution for Two Populations**

Use a difference sampling distribution if the question presents 2 different populations.

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**Sampling Distribution for Two Populations: Example**

(adapted from AP Statistics – Chapter 9 – Sampling Distribution Multiple Choice Questions Medium oranges have a mean weight of 14oz and a standard deviation of 2oz. Large oranges have a mean weight of 18oz and a standard deviation of 3oz. Find the probability of finding a medium orange that weights more than a large orange.

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Example Problem (adapted from DeVeau Sampling Distribution Models Exercise #42) Ayrshire cows average 47 pounds if milk a day, with a standard deviation of 6 pounds. For Jersey cows, the mean daily production is 43 pounds, with a standard deviation of 5 pounds. Assume that Normal models describe milk production for these breeds. A) We select an Ayrshire at random. What’s the probability that she averages more than 50 pounds of milk a day? B) What’s the probability that a randomly selected Ayrshire gives more milk than a randomly selected Jersey? C) A farmer has 20 Jerseys. What’s the probability that the average production for this small herd exceeds 45 pounds of milk a day? D) A neighboring farmer has 10 Ayrshires. What’s the probability that his herd average is at least 5 pounds higher than the average for the Jersey herd?

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**Example Problem Solution**

First, check the assumptions: Independent samples Randomness Sample represents less than 10% of population

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**Example Problem Solution**

A) Use the normal model to estimate the appropriate probability.

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**Example Problem Solution**

B) Create a normal model for the difference between Ayrshires and Jerseys. Use the model to estimate the appropriate probability.

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**Example Problem Solution**

C) Create a sampling distribution model for which n=20 Jerseys. Use the model to estimate the appropriate probability.

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**Example Problem Solution**

D) First create a sampling distribution model for 10 random Ayrshires and 20 random Jerseys. Then create a normal model for the difference between the 10 Ayrshires and 20 Jerseys.

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