AP Statistics: Section 9.1 Sampling Distributions
What is the usual way to gain information about some characteristic of a population?
We must note, however, that the sample information we gather may differ from the true population characteristic we are trying to measure. Furthermore, the sample information may differ from sample to sample.
This sample-to-sample variability, called ____________________________ poses a problem when we try to generalize our findings to the population. We need to gain an understanding of this variability.
A parameter is A statistic is
In statistical practice, the value of a parameter is unknown since we cannot examine the entire population. In practice, we often use a statistic to estimate an unknown parameter.
The population mean is represented by the symbol ___ (Greek: Mu), the population standard deviation by ___(Greek: Sigma) and the population proportion by ___.
The sample mean is represented by the symbol ____ (x bar), the sample standard deviation by ____ and the sample population by ____ (p hat).
Example: Identify the number that appears in boldface type as a parameter or a statistic, and then write an equation using the proper symbol from above and the number from the statement
A department store reports that 84% of all customers who use the store’s credit plan pay their bills on time.
A consumer group, after testing 100 batteries of a certain brand, reported an average of 63 hr of use.
We can view a sample statistic as a random variable, because we have no way of predicting exactly what statistic value we will get from a sample, BUT, given a population parameter, we know how these sample statistics will behave in repeated sampling.
Before we continue, we need to discuss two important definitions:
The population distribution of a variable is the distribution of values of the variable among all individuals in the population.
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 population.
Careful: The population distribution describes the individuals that make up the population. A sampling distribution describes how a statistic varies in many samples of size n from the population.
Consider flipping a coin 10 times. We would expect to get 5 heads out of the 10, but we realize that we could also get 4 or 6 or 7 or even 10. Let’s simulate this using our graphing calculators.
How many different samples of size 10 are possible in this situation?
Let’s increase our sample size to 25.
Hopefully, most of us found that as we increased the sample size from 10 to 25, the mean and the median of our sample proportions became closer together and both became closer to.5. Also we should find that the standard deviation grows smaller and our distribution of the sample proportions became closer to being a normal distribution.
Since a sampling distribution is a distribution, we can use the tools of data analysis to describe the distribution: ________, ________, __________ and __________.
Example: According to 2005 Nielsen ratings, Survivor: Guatemala was one of the most- watched TV shows in the US during every week that it aired. Suppose that the true proportion of US adults who watched Survivor: Guatemala was p = 0.37.
Describe the distribution of sample proportions at the right for samples of size n = 100 of people who watched Survivor: Guatemala.
Describe the distribution of sample proportions for samples of size n = 1000 of people who watched Survivor: Guatemala.
A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution equals the true value of the population parameter. The statistic is called an unbiased estimator of the parameter.
An unbiased statistic will sometimes fall above the true value of the parameter and sometimes below. There is no tendency to overestimate or underestimate the parameter, hence the “unbiased.”
We will see in sections 9.2 & 9.3 that are both unbiased estimators of population parameters.
The variability of a statistic is described by the spread of its sampling distribution.
This spread is determined by the sampling design and the sample size. Larger samples give a ________ spread.
As long as the population is larger than the sample by at least a factor of 10, the spread of the sampling distribution is approximately the same for any population size.
This means that a statistic from an SRS of size 2500 from the more than 300 million residents of the US is just as precise as an SRS of size 2500 from the 750,000 inhabitants of San Francisco.
For a better understanding of bias and variability, think of the center of a target as the true population parameter and an arrow shot at the target as a sample statistic.