Sampling Distribution and Central Limit Theorem Notes

Sampling Distribution Models

Candy Sampling Activity

• If still needing to sample candy, do so immediately upon entering the classroom.
  • $n = 5/25/50$ (sample sizes).
  • Proportion of blue candies.
• On the back whiteboard, make a tick mark in the appropriate row that corresponds to the value of the proportion of blue candies for each of the three sample sizes.
  • If the category lists a range of values, the range is inclusive of the left endpoint and exclusive of the right endpoint.
  • For example: 0.3-0.4 would include the value of 0.3, but not include the value of 0.4.

Chapter 17

• Sampling Distribution Models.
• Don't forget to read this chapter!

Samples and Descriptive Statistics

• We have talked about samples and created descriptive statistics.
  • Sample proportion for categorical variables.
  • Sample mean for quantitative variables.
• These statistics vary from sample to sample and each is an estimate of a population model parameter.
• We have discussed the notion of sampling error (or sampling variability) and understand that it is unavoidable.
• Sampling error is understandable and predictable.
• We hope it is small, but we can know how large it is likely to be.

Using Sample Results

• Using the knowledge on the previous slides, we can look at sample results and reach important decisions.
• We may decide that the results are within a reasonable margin of variation from what we expected, OR…
• We may deem our results to be so unlikely that we do not believe they could simply be sampling error (making our outcome statistically significant).
• We are about to begin the process of learning how to make that distinction.

Transitioning Focus

• We used to focus on the data and derive a statistic from it.
• Now, we focus on the statistic itself.
• Population -> Sampling Distribution of sample proportions.
  • Mean of the distribution = $p$.
  • Standard Deviation of the distribution = ?

Modeling the Sampling Distribution of Sample Proportions

• We need to be able to understand the characteristics of the sampling distribution WITHOUT repeated sampling.
• We COULD simulate repeated sampling, OR we could simply use the patterns that have emerged via those simulations.
• It turns out that the histogram through repeated sampling is unimodal, symmetric, and centered at p.
• More specifically, it is a fortunate fact that a Normal model is just the right one for the histogram of sample proportions.

Modeling the Distribution of Sample Proportions (cont.)

• A sampling distribution model for how a sample proportion varies from sample to sample allows us to quantify that variation and how likely it is that we’d observe a sample proportion in any particular interval.
• What information do we need to fully describe a distribution?
  • Center.
  • Variability.
  • Shape.
• Our "model" of the distribution of sample proportions is Normal, with a mean of p, and a standard deviation that follows a predictable pattern.

Modeling the Distribution of Sample Proportions (cont.)

• When working with proportions, knowing the mean automatically gives us the standard deviation as well—the standard deviation we will use is
• $\sqrt{\frac{p(1-p)}{n}}$
• So, the distribution of the sample proportions is modeled with a probability model that is normal.

Picture of Sampling Distribution

• A picture of what we just discussed is as follows:
• So we should not be surprised if 95% of various polls gave results that were near the mean but varied above and below that by no more than two standard deviations.
• This is sampling error or sampling variability.

Assumptions and Conditions

• Most models are useful only when specific assumptions are true.
• There are two assumptions in the case of the model for the distribution of sample proportions:
  • The Independence Assumption: The sampled values must be independent of each other.
  • The Sample Size Assumption: The sample size, n, must be large enough.
• We may not be able to know if an assumption is true, but we can check certain conditions that, if met, make the assumption a reasonable one to make.

Assumptions and Conditions (cont.)

• Under the Independence Assumption, we can check the following conditions:
  1. Random sample or random assignment:
     • If a sample was taken, it should ideally be a simple random sample of the population.
     • Check for biases and make sure the sample was representative of the population.
     • If an experiment was conducted, the subjects should have been randomly assigned to the treatments.
  2. Population > 10n:
     • The sample size, n, must be no larger than 10% of the population.
     • Often we have a very large population of an unknown size.
     • It is most important to think about this condition when you have a relatively small population of a known size.

Assumptions and Conditions (cont.)

• Under the Sample Size Assumption, we can check the following condition:
  1. $np ≥ 10$ and $n(1-p) ≥ 10$:
     • The sample size has to be big enough so that both np (number of successes) and n(1-p) (number of failures) are at least 10.

Sampling Distribution Models - Day 2

• Chapter 17.

In Summary

• A proportion is no longer just a computation from a set of data.
• It is now a random variable quantity that has a probability distribution.
• This distribution is called the sampling distribution model for proportions.
• Even though we depend on sampling distribution models, we never actually get to see them.
• We never actually take repeated samples from the same population and make a histogram. We only imagine or simulate them.

Summary (cont.)

• Still, sampling distribution models are important because:
  • they act as a bridge from the real world of data to the imaginary world of the statistic and
  • enable us to say something about the population when all we have is data from the real world.

Summary (cont.)

• Provided that the sampled values are independent and the sample size is large enough (meaning the conditions are met), the sampling distribution of$\hat{p}$ is modeled by a Normal model with
  • Mean: $p$
  • Standard deviation: $\sqrt{\frac{p(1-p)}{n}}$

Example Worksheet

• Problem 1

How Sample Size Affects Model

• We know that a Normal model becomes more useful as sample size increases.
• Another general concept: As sample size increases, variability decreases.
• Let's check our M&M data for n=5, n=25, and n=50.

Quantitative Data

• Proportions summarize categorical variables.
• The Normal sampling distribution model looks like it will be very useful.
• Can we do something similar with quantitative data?
• We can. Not only can we use all the same concepts, but almost the same model.
• Like any statistic computed from a random sample, a sample mean also has a sampling distribution.
• We can use simulation to get a sense as to what the sampling distribution of the sample mean might look like…

Means – The "Average" of One Die

• Let's start with a simulation of 10,000 tosses of a die. A histogram of the results is:

Means – Averaging More Dice

• Looking at the average of two dice after a simulation of 10,000 tosses:
• The average of three dice after a simulation of 10,000 tosses looks like:

Means – Averaging Still More Dice

• The average of 5 dice after a simulation of 10,000 tosses looks like:
• The average of 20 dice after a simulation of 10,000 tosses looks like:

Means – What the Simulations Show

• As the sample size (number of dice) gets larger, each sample average is more likely to be closer to the population mean.
• So, we see the shape continuing to tighten around 3.5
• And, it should not surprise you that the sampling distribution of a mean becomes Normal.

The Fundamental Theorem of Statistics

• The sampling distribution of any mean becomes more nearly Normal as the sample size grows, regardless of the shape of the population distribution.
• The Fundamental Theorem of Statistics is called the Central Limit Theorem (CLT).

The Central Limit Theorem (CLT)

• The mean of a random sample is a random variable whose sampling distribution can be approximated by a Normal model.
• The larger the sample, the better the approximation will be.

Assumptions and Conditions for CLT

• The CLT requires essentially the same assumptions we saw for modeling proportions:
  • Independence Assumption: The sampled values must be independent of each other.
  • Sample Size Assumption: The sample size must be sufficiently large.

Assumptions and Conditions (cont.) for CLT

• To make our assumptions plausible, we need to check the following conditions:
  • Random sample or Random Assignment: The data values must be sampled or assigned randomly.
  • Population > 10n: When the sample is drawn without replacement, the sample size, n, should be no more than 10% of the population.
  • $n ≥ 30$ or ????: The CLT doesn't tell us how large a sample we need. Different sources use different minimums. We will use.
  • But, there is another way this condition could be met.

Large Enough Sample Condition

• With a small sample size, there still is a possibility that the distribution of sample means may be approximately Normal.
• When the sample size is small, the sampling distribution takes on the characteristics of the population from which it was chosen.
• If the population is skewed to the right, the sampling distribution will be skewed to the right as well with small sample sizes, and so on.
• Therefore, if the population is known to be Normal, then the distribution of sample means chosen from that population will be Normal as well.

Modeling the Distribution of Sample Means (cont.)

• Our "model" of the distribution of sample means is Normal, with a mean of μ, and a standard deviation that follows a predictable pattern.
• The standard deviation we will use is $\frac{\sigma}{\sqrt{n}}$
• So, the distribution of the sample means is modeled with a probability model that is Normal.

Example Worksheet

• Problem 2

Sampling Distribution Models - Day 3

• Chapter 17

Comparison of Conditions

Distribution of Sample Proportions

• Random Sample from population or Random Assignment to groups.
• Population size > 10n.
• $np ≥ 10$ and $n(1-p) ≥ 10$.

Distribution of Sample Means

• Random Sample from population or Random Assignment to groups.
• Population size > 10n.
• $n ≥ 30$ or population is Normal.

Comparison of Sampling Distributions

Distribution of Sample Proportions

• The sampling distribution has a Normal model.

Distribution of Sample Means

• The sampling distribution has a Normal model.

About Variation

• The standard deviation of the sampling distribution declines only with the square root of the sample size (the denominator contains the square root of n).
• Therefore, the variability decreases as the sample size increases.
• While we’d always like a larger sample, the square root limits how much we can make a sample tell about the population.

Real World vs Model World

• Be careful! Now we have two distributions to deal with.
  • The first is the real-world distribution of the sample, which we might display with a histogram.
  • The second is the math world sampling distribution of the statistic, which we model with a Normal model based on the Central Limit Theorem.
• Don’t confuse the two!
• “Normal” vs. “approximately” Normal

Sampling Distribution Models

• Always remember that the statistic itself is a random quantity.
• We can’t know what our statistic will be because it comes from a random sample.
• Fortunately, for the mean and proportion, the CLT tells us that we can model their sampling distribution directly with a Normal model.
• There are two basic truths about sampling distributions:
  1. Sampling distributions arise because samples vary. Each random sample will have different cases and, so, a different value of the statistic.
  2. Although we can always simulate a sampling distribution, the Central Limit Theorem saves us the trouble for means and proportions.
• Our parameter does not vary. It is a true value about the population that is not changing. Often we never know the value of a parameter.

Example Worksheet

• Problems 3-6