Sampling Distribution Models Population – all items of interest. Population Parameter: p Inference We are ready to start putting together the pieces to come up with how to use sample statistics to make inferences, educated guesses, about population parameters. Remember our picture of the population and sample. We will not complete the process by introducing the idea of using a sample statistics as an estimate of the population parameter. But before that we need to know a little more about how sample statistics based on random samples from a population behave. Sample – a few items from the population. Sample Statistic: Random selection

Sampling Distribution of Shape: Approximately Normal Center: The mean is p. Spread: The standard deviation is

Sampling Distribution of Conditions: 10% Condition: The size of the sample should be less than 10% of the size of the population. Success/Failure Condition: np and n(1 – p) should both be greater than 10.

68 – 95 – 99.7 Rule

Probability If the population proportion, p, is known, we can find the probability or chance that takes on certain values using a normal model.

Inference In practice the population parameter, p, is not known and we would like to use a sample to tell us something about p. Use the sample proportion, , to make inferences about the population proportion p.

Example Population: All adults in the U.S. Parameter: Proportion of all adults in the U.S. who think that abortion should be legal. Unknown!

Example Sample: 1,772 randomly selected registered voters nationwide. Quinnipiac University Poll, Jan. 30 – Feb. 4, 2013. Statistic: 992 of the 1,772 registered voters in the sample (56%) answered that abortion should be legal.

68-95-99.7 Rule 95% of the time the sample proportion, , will be between

68-95-99.7 Rule 95% of the time the sample proportion, , will be within two standard deviations of p.

Standard Deviation Because p, the population proportion is not known, the standard deviation is also unknown.

Standard Error Substitute as our estimate (best guess) of p. The standard error of is:

About 95% of the time the sample proportion, , will be within two standard errors of p.

About 95% of the time the population proportion, p, will be within two standard errors of .

Confidence Interval for p We are 95% confident that p will fall between

Example

Interpretation We are 95% confident that the population proportion of all adults in the U.S. who would answer that abortion should be legal is between 53.6% and 58.4%.

Interpretation Plausible values for the population parameter p. 95% confidence in the process that produced this interval.

95% Confidence If one were to repeatedly sample at random 1,772 adults and compute a 95% confidence interval for each sample, 95% of the intervals produced would contain, or capture, the population proportion p.

Simulation http://statweb.calpoly.edu/chance/applets/Confsim/Confsim.html

Margin of Error Is called the Margin of Error (ME). This is the furthest can be from p, with 95% confidence.

Margin of Error What if we want to be 99.7% confident?

Margin of Error Confidence z* 80% 90% 95% 98% 99% 1.282 1.645 2.326 2.576

Another Example Pew Research Center/USA Today Poll, Feb. 13 – 18, 2013. Asked of 1,504 randomly selected adults nationwide. “Do you favor or oppose setting stricter emission limits on power plants in order to address climate change?”

Another Example n=1,504 randomly selected adults. Favor Oppose Unsure/ Refused 52% 28% 10%

Another Example 90% confidence interval for p, the proportion of the population of all adults in the U.S. who favor emission limits on power plants in order to address climate change.

Calculation

What Sample Size? Conservative Formula The sample size to be 95% confident that , the sample proportion, will be within ME of the population proportion, p.

Example Suppose we want to be 95% confident that our sample proportion will be within 0.02 of the population proportion.