Presentation on theme: "Sampling Distributions"— Presentation transcript:
1 Sampling Distributions A review by Hieu Nguyen (03/27/06)
2 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
3 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
4 BiasThe 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
5 Sampling VariabilitySamples 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.
6 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.
7 CLT ExampleImagine 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.
8 CLT ExamplePlot 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
9 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.
10 Two Most Common Types of Sampling Distributions Sample Proportion DistributionDistribution of the sample proportions of samples from a populationSample Mean DistributionDistribution of the sample means of samples from a populationFor both types, the ideal shape is a normal distribution
11 Sampling Distributions: Conditions Before assuming that a sampling distribution is normal, check the following conditions:Plausible IndependenceRandomnessEach sample is less than 10% of the population
12 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
13 Sampling Distributions As Normal Distributions: Equations Sample Proportion Distributionp = population proportion (given)Sample Mean Distributionμ = population mean (given)σ = population standard deviation (given)
14 Sampling Distributions As Normal Distributions: Note Note: If any of the parameters are unknown, use the statistics from a sample to approximate it.
15 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.
16 Using Sampling Distributions: Z-Scores w/ Example Use the z-score table to find appropriate probabilitiesExample: Find the probability that a poll of Americans that support Alito’s nomination will return a sample proportion of .72.
17 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.
18 Sampling Distribution for Two Populations Use a difference sampling distribution if the question presents 2 different populations.
19 Sampling Distribution for Two Populations: Example (adapted from AP Statistics – Chapter 9 – Sampling Distribution Multiple Choice QuestionsMedium 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.
20 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?
21 Example Problem Solution First, check the assumptions:Independent samplesRandomnessSample represents less than 10% of population
22 Example Problem Solution A) Use the normal model to estimate the appropriate probability.
23 Example Problem Solution B) Create a normal model for the difference between Ayrshires and Jerseys. Use the model to estimate the appropriate probability.
24 Example Problem Solution C) Create a sampling distribution model for which n=20 Jerseys. Use the model to estimate the appropriate probability.
25 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.