 Normal Curves  The family of normal curves  The rule of  The Central Limit Theorem  Confidence Intervals  Around a Mean  Around a Proportion

 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.

 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

 The 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

 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

 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.

 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

 The 95% confidence interval around a sample mean is:

 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.

 The 95% confidence interval around a sample proportion is: And the 99.7% confidence interval would be:

 The margin of error is calculated by:

 In a poll of 505 likely voters, the Field Poll found 55% support for a constitutional convention.

 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.