Sampling Distributions Chapter 18

Sampling Distributions If we could take every possible sample of the same size (n) from a population, we would create the sampling distribution. Try to picture taking infinite samples of size n from a population.

Sampling Distributions Why do we sample? Averages and proportions are less variable and more normal than individual observations

Sampling Distributions Categorical data produces distributions that are based on proportions.

Sampling Distributions A parameter is a measure of the population. This value is typically unknown. For proportions this is p or π A statistic is a measure from a sample. We often use a statistic to estimate an unknown parameter. For proportions this is .

Sampling Distributions Sampling variability: in repeated random sampling, the value of the statistic will vary. (We don’t expect the same value every time we sample do we!)

Sampling Distributions We expect variability when we sample. Because of this we create a set of values that fall around the center of the distribution, p. This variability creates a curve that under the right conditions will be approximately normal.

Sampling Distributions When we create a distribution using repeated samples from categorical data, this is known as the sampling distribution for a proportion.

Sampling Distributions When we describe a distribution we want to focus on Shape, Center, and Spread (Remember CUSS & BS?)

Sampling Distributions of Proportions In order to use a normal model for sampling distributions of proportions and the formulas that follow, the following 3 conditions must be met. Each condition has it’s own purpose. You should know why you need each one.

Sampling Distributions of Proportions Conditions: 1) Randomization. This helps insure that your data was fairly collected and not biased in some way. You need to state that your data comes from an SRS, was fairly collected, was randomly chosen etc…

Sampling Distributions of Proportions Conditions: 2) Independence. This protects our standard deviation formula and keeps it accurate. We must insure (we usually assume) that the sampled values are independent of one another. If we are sampling without replacement, then we must state that our sample is no more than 10% of our population.

Sampling Distributions of Proportions Conditions: 3) Large enough sample. To insure that the sample size is large enough to approximate normal, we must expect at least 10 successes and at least 10 failures. np  10 and n(1 – p)  10

Sampling Distributions of Proportions Provided conditions are met, the sampling distribution of a proportion will be normal with mean p and standard deviation Or in notation N(p, )

Sampling Distributions for Means When the data is quantitative, your distribution is based on repeated averages from the samples.

Sampling Distributions for Means A parameter is a measure of the population. This value is typically unknown. For means this is μ. A statistic is a measure from a sample. We often use a statistic to estimate an unknown parameter. For means this is .

Sampling Distributions of Sample Means When we create a distribution using every possible sample of a given size from quantitative data, this is known as the sampling distribution for a mean.

Sampling Distributions of Sample Means The shape of the sampling distribution depends on the shape of the population it is drawn from. ** If the population is normal, then the distribution of the sample mean will be normal (regardless of sample size).

Sampling Distributions of Sample Means The shape of the sampling distribution depends on the shape of the population. **For skewed or odd shaped distributions, if the sample size is large enough, the sampling distribution will be approximately normal. So…how large is large enough?

Sampling Distributions of Sample Means The Central Limit Theorem (CLT) CLT addresses two things in a distribution, shape and spread. As the sample size increases: The shape of the sampling distribution becomes more normal The variability of the sampling distribution decreases

Sampling Distributions of Sample Means The Law of Large Numbers Draw observations at random from any population with given mean . As n increases, the sample mean gets closer and closer to the true population mean, .

CLT vs LLN!! CLT - focuses on shape and spread Law of Large Numbers – focuses on center

Sampling Distributions of Sample Means Conditions: Randomization. Independence. (Same as the first two conditions for Proportions)

Sampling Distributions of Sample Means Conditions: 3) Large Enough Sample. There is no “for sure” way to tell if your sample is large enough. It is common practice that if your sample is at least 30 (n ≥ 30), you are OK to assume normal for the sampling distribution. (Remember, if the distribution is given normal, then any sample size is OK)

Sampling Distributions of Sample Means When conditions are met, and the data is quantitative, the sampling distribution is normal with a center at the population mean, μ, and a standard deviation at So…. N( , )

Sampling Distributions We said at the beginning that in most real life cases, we will not know the population parameters (µ, σ, p or π) so we will have to use the sample statistics as estimates of those. Our terminology changes just a little…

Sampling Distributions  

Sampling Distributions  

Sampling Distributions  

Adjusting Sample Size Questions about sample size often come up. If we want to reduce variability one thing we can do is increase sample size. Sometimes we must figure out how much standard deviation we can have, then determine what sample size will get us there. We can use the formulas for standard deviation and solve for sample size.

Adjusting Sample Size Shortcut!!! Since the standard deviation decreases at a rate of √n, taking a sample 4 times as large reduces the standard deviation by ½.