7.1 What is a Sampling Distribution?

ObjectivesSWBAT:

• DISTINGUISH between a parameter and a statistic.• USE the sampling distribution of a statistic to EVALUATE a claim about a parameter.• DISTINGUISH among the distribution of a population, the distribution of a sample, and

the sampling distribution of a statistic.• DETERMINE whether or not a statistic is an unbiased estimator of a population

parameter.• DESCRIBE the relationship between sample size and the variability of a statistic.

What is a parameter? What is a statistic? How is one related to the other?• The process of statistical inference involves using information from a sample to draw

conclusions about a wider population.• As we begin to use sample data to draw conclusions about a wider population, we must be

clear about whether a number describes a sample or a population.

A parameter is a number that describes some characteristic of the population. A statistic is a number that describes some characteristic of a sample.

Remember s and p: statistics come from samples and

parameters come from populations

Note: Be aware! Confusing these ideas is a common source of lost credit on the AP exam.

Example: Identify the population, the parameter, the sample, and the statistic.a) A pediatrician wants to know the 75th percentile for the distribution of heights of 10-

year-old boys, so she takes a sample of 50 patients and calculates Q3 = 56 inches.Population: all 10-year-old boys Parameter: the 75th percentile for all 10-year-old boys Sample: 50 10-year-old boys included in the sample Statistic: 56 inches, the 75th percentile of the heights in the sample b) A Pew Research Center Poll asked 1102 12- to 17-year-olds in the US if they have a cell phone. Of the respondents, 71% said “Yes.”Population: all 12- to 17-year-olds in the United StatesParameter: p, the proportion of all 12- to 17-year olds with cell phonesSample: 1102 12- to 17-year-olds in the sample Statistic: the sample proportion with a cell phone, = 0.71

What is sampling variability? Why do we care?

This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling.To make sense of sampling variability, we ask, “What would happen if we took many samples?”

The sample mean and sample proportions will vary, but the parameter will remain constant.We need to estimate sampling variability so we know how close our estimates are to the truth (margin of error: more to come in Chapter 8.

What is a sampling distribution? Why do we care?• The definition of a “distribution” is always the same: it describes the possible values and

how often the values occur. In this case, a sampling distribution is a distribution of a statistic.

• If we took every one of the possible samples of size n from a population, calculated the sample proportion for each, and graphed all of those values, we’d have a sampling distribution.

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

• We care because we need to know what counts as unusual.

What is the difference between the distribution of the population, the distribution of the sample, and the sampling distribution of a sample statistic?

Our population is 200 chips, 100 of which are red and 100 of which are blue. The population proportion of red chips is p = 0.5.Suppose we choose 500 SRS’s of size n = 20 from the population of 200 chips.The population is on the left.Each of the 500 SRS’s are in the middle, along with their accompanying ’s.The distribution of all the ’s from the SRS’s is at the right.

• The population distribution gives the value of the variable for all individuals in the population.

• The distribution of sample data shows the values of the variable for the individuals in the sample.

• The sampling distribution collects the values of the statistics from all possible samples of the same size and displays these values.

• Note: You can run into danger if you make ambiguous statements on the AP test. For example, by saying “when the sample size increases, the variability decreases.” You need to specify the variability of which distribution! Always specify which distribution you are talking about.

What is an unbiased estimator? What is a biased estimator? Provide some examples.• The fact that statistics from random samples have definite sampling distributions allows us to

answer the question, “How trustworthy is a statistic as an estimator of the parameter?”• In the chips example, we collected many samples of size 20 and calculated the sample

proportion of red chips. How well does the sample proportion estimate the true proportion of red chips, p = 0.5?

A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

Note that the center of the approximate sampling distribution is close to 0.5. In fact, if we took ALL possible samples of size 20 and found the mean of those sample proportions, we’d get exactly 0.5.

• A biased estimator has a mean of its sampling distribution not equal to the true value of the parameter being estimated.

• Unbiased doesn’t mean perfect! Unbiased means not consistently too high or consistently too low when taking many random samples. Biased means the statistic is consistently higher or lower than the parameter.

• If the sampling process is biased (undercoverage, response, non-response) there are no guarantees, even if a statistic is an unbiased estimator.

• As we saw before, the chips example is an example of an unbiased estimator.• Think back to Chapter 4 for an example of a biased estimator. Remember the

example trying to find the proportion of town residents that opposed budget cuts to the high school’s athletic funding. The person conducted a convenience sample by sampling people at the football game. Clearly this is a biased estimator as it would consistently be higher than the population proportion.

How can you reduce variability of a statistic?• Take a larger sample. n=100 n=1000

• Above are two sampling distributions, both attempting to find the proportion of people that watch the show Survivor. Suppose the true proportion of US adults who have watched Survivor is p = 0.37.

• The graph to the left drew 400 SRS’s of size n = 100. We see that a sample of 100 people often gave a sample proportion quite far from the population parameter.

• The graph to the right drew 400 SRS’s of size n = 1000. The sample of 1000 people gave a sample proportion much closer to the population parameter.

• Another way to reduce variability is by a better design, such as stratified sampling.

What is the difference between accuracy and precision?Accurate = unbiased

Precise = low variability