Normal Distribution

STAT 101 Dr. Kari Lock Morgan 10/18/12 Normal Distribution • Chapter 5 • Normal distribution • Central limit theorem • Normal distribution for confidence intervals • Normal distribution for p-values • Standard normal

Project 1 • Due Tuesday • 5 pages, double spaced, including figures • Hypotheses should not change based on data • This is a research paper – there should be text and complete sentences.

Bootstrap and Randomization Distributions Correlation: Malevolent uniforms Slope :Restaurant tips All bell-shaped distributions! What do you notice? Mean :Body Temperatures Diff means: Finger taps Proportion : Owners/dogs Mean : Atlanta commutes

Normal Distribution • The symmetric, bell-shaped curve we have seen for almost all of our bootstrap and randomization distributions is called a normal distribution

Central Limit Theorem! For a sufficiently large sample size, the distribution of sample statistics for a mean or a proportion is normal www.lock5stat.com/StatKey

Central Limit Theorem • The central limit theorem holds for ANY original distribution, although “sufficiently large sample size” varies • The more skewed the original distribution is (the farther from normal), the larger the sample size has to be for the CLT to work

Central Limit Theorem • For distributions of a quantitative variable that are not very skewed and without large outliers, n ≥ 30 is usually sufficient to use the CLT • For distributions of a categorical variable, counts of at least 10 within each category is usually sufficient to use the CLT

Normal Distribution • The normal distribution is fully characterized by it’s mean and standard deviation

Normal Distribution

Bootstrap Distributions If a bootstrap distribution is approximately normally distributed, we can write it as N(parameter, sd) N(statistic, sd) N(parameter, se) N(statistic, se) sd = standard deviation of variable se = standard error = standard deviation of statistic

Confidence Intervals If the bootstrap distribution is normal: To find a P% confidence interval , we just need to find the middle P% of the distribution N(statistic, SE)

Best Picture What proportion of visitors to www.naplesnews.comthought The Artist should win best picture?

Best Picture www.lock5stat.com/statkey

Area under a Curve • The area under the curve of a normal distribution is equal to the proportion of the distribution falling within that range • Knowing just the mean and standard deviation of a normal distribution allows you to calculate areas in the tails and percentiles www.lock5stat.com/statkey

Best Picture www.lock5stat.com/statkey

Best Picture

Confidence Intervals For a normal sampling distribution, we can also use the formula to give a 95% confidence interval.

Confidence Intervals For normal bootstrap distributions, the formula gives a 95% confidence interval. How would you use the N(0,1) normal distribution to find the appropriate multiplier for other levels of confidence?

Confidence Intervals For a P% confidence interval, use where P% of a N(0,1) distribution is between –z* and z*

Confidence Intervals P% -z* z*

Confidence Intervals Find z* for a 99% confidence interval. www.lock5stat.com/statkey z* = 2.575

News Sources “A new national survey shows that the majority (64%) of American adults use at least three different types of media every week to get news and information about their local community” The standard error for this statistic is 1% Find a 99% confidence interval for the true proportion. Source: http://pewresearch.org/databank/dailynumber/?NumberID=1331

News Sources

Confidence Interval Formula From N(0,1) From original data From bootstrap distribution

First Born Children • Are first born children actually smarter? • Based on data from last semester’s class survey, we’ll test whether first born children score significantly higher on the SAT • From a randomization distribution, we find SE = 37

First Born Children What normal distribution should we use to find the p-value? N(30.26, 37) N(37, 30.26) N(0, 37) N(0, 30.26) Because this is a hypothesis test, we want to see what would happen if the null were true, so the distribution should be centered around the null. The variability is equal to the standard error.

Hypothesis Testing

p-values If the randomization distribution is normal: To calculate a p-value, we just need to find the area in the appropriate tail(s) beyond the observed statistic of the distribution N(null value, SE)

First Born Children N(0, 37) www.lock5stat.com/statkey p-value = 0.207

First Born Children

Standard Normal • Sometimes, it is easier to just use one normal distribution to do inference • The standard normal distribution is the normal distribution with mean 0 and standard deviation 1

Standardized Test Statistic • The standardized test statistic is the number of standard errors a statistic is from the null value • The standardized test statistic (also called a z-statistic) is compared to N(0,1)

p-value Find the standardized test statistic: The p-value is the area in the tail(s) beyond z for a standard normal distribution

First Born Children Find the standardized test statistic

First Born Children Find the area in the tail(s) beyond z for a standard normal distribution p-value = 0.207

z-statistic If z = –3, using  = 0.05 we would (a) Reject the null (b) Not reject the null (c) Impossible to tell (d) I have no idea About 95% of z-statistics are within -2 and +2, so anything beyond those values will be in the most extreme 5%, or equivalently will give a p-value less than 0.05.

z-statistic • Calculating the number of standard errors a statistic is from the null value allows us to assess extremity on a common scale

Formula for p-values From original data From H0 From randomization distribution Compare z to N(0,1) for p-value

Standard Error • Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? • We can!!! • Or rather, we’ll be able to next week!

To Do • Do Project 1 (due 10/23) • Read Chapter 5

Normal Distribution