Discrete distribution word problems –Probabilities: specific values, >, =, … –Means, variances Computing normal probabilities and “inverse” values: –Pr(X<y) when y is above and below the mean of X –Pr(y 1 <X<y 2 ) when y 1 and y 2 are: both above the mean of X both below the mean of X on opposite sides of the mean of X Central Limit Theorem: –Sum version –Average version

Apply Central Limit Theorem to Estimates of Proportions Source: gallup.com Suppose this is based on a poll of 100 people

This uses the “average” version of the CLT. Two lectures ago, we applied the “sum” version of the CLT to the binomial distribution.

Suppose true p is If survey is conducted again on 49 people, what’s the probability of seeing 38% to 42% favorable responses? Pr( 0.38 < P < 0.42) = Pr[( )/sqrt(0.62*0.38/49) < Z < ( )/sqrt(0.62*0.38/49) ] = Pr(-0.29 < Z < 0.29) = 2*Pr(Z<-0.29) = 0.77

Chapter 8: In the previous example, the random quantity was the estimator. Examples of estimators: Sample mean = X = (X 1 +…+X n )/n Sample variance = [(X 1 -X) 2 +…+(X n -X) 2 ]/(n-1) Sample median = midpoint of the data… Regression line = …. ESTIMATORS CALCULATE STATISTISTICS FROM DATA If data are random, then the estimators are random too.

Central limit theorem tells us that the estimators X and P have normal distributions as n gets large: X ~ N( ,  2 /n) where  and  are the mean and standard deviation of the random variables that go into X. P ~ N(p,p(1-p)/n) where p is true proportion of “yeses”

Two ways of ways to evaluate estimators: –Bias: “Collect the same size data set over and over. Difference between the average of the estimator and the true value is the bias of the estimator.” –Variance: Collect the same size data set over and over. Variability is a measure of how closely each estimate agrees.

True value Distribution of an unbiased estimator Distribution of a biased estimator Bias = inaccuarcy Variance = imprecision True value Distribution of a less variable estimator Distribution of a more variable estimator Example: The median is a biased estimate of the true mean when the distribution is skewed.