Political Science 30: Political Inquiry

1. Political Science 30:Political Inquiry

2. The Magic of the Normal Curve • Normal Curves • The family of normal curves • The rule of 68-95-99.7 • The Central Limit Theorem • Confidence Intervals • Around a Mean • Around a Proportion

3. Normal Curves (or distributions) • Normal curves are a special family of density curves, which are graphs that answer the question: “what proportion of my cases take on values that fall within a certain range?” • Many things in nature, such as sizes of animals and errors in astronomic calculations, happen to be normally distributed.

4. Normal Curves

5. Normal Curves • What do all normal curves have in common? • Symmetric • Mean = Median • Bell-shaped, with most of their density in center and little in the tails • How can we tell one normal curve from another? • Mean tells you where it is centered • Standard deviation tells you how thick or narrow the curve will be

6. Normal Curves • The 68-95-99.7 Rule. • 68% of cases will take on a value that is plus or minus one standard deviation of the mean • 95% of cases will take on a value that is plus or minus two standard deviations • 99.7% of cases will take on a value that is plus or minus three standard deviations

7. The Central Limit Theorem • If we take repeated samples from a population, the sample means will be (approximately) normally distributed. • The mean of the “sampling distribution” will equal the true population mean. • The “standard error” (the standard deviation of the sampling distribution) equals

8. The Central Limit Theorem • A “sampling distribution” of a statistic tells us what values the statistic takes in repeated samples from the same population and how often it takes them.

9. Confidence Intervals • We use the statistical properties of a distribution of many samples to see how confident we are that a sample statistic is close to the population parameter • We can compute a confidence interval around a sample mean or a proportion • We can pick how confident we want to be • Usually choose 95%, or two standard errors

10. Confidence Intervals • The 95% confidence interval around a sample mean is:

11. Confidence Intervals • If my sample of 100 donors finds a mean contribution level of $15,600 and I compute a confidence interval that is: $15,600 + or - $600 • I can make the statement: I can say at the 95% confidence level that the mean contribution for all donors is between $15,000 and $16,200.

12. Confidence Intervals • The 95% confidence interval around a sample proportion is: And the 99.7% confidence interval would be:

13. What Determines the “Margin of Error” of a Poll? • The margin of error is calculated by:

14. What Determines the “Margin of Error” of a Poll? • In a poll of 505 likely voters, the Field Poll found 55% support for a constitutional convention.

15. What Determines the “Margin of Error” of a Poll? • The margin of error for this poll was plus or minus 4.4 percentage points. • This means that if we took many samples using the Field Poll’s methods, 95% of the samples would yield a statistic within plus or minus 4.4 percentage points of the true population parameter.