1. 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

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

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

4. Suppose true p is 0.40. 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[(0.38-0.40)/sqrt(0.62*0.38/49) < Z < (0.42-0.40)/sqrt(0.62*0.38/49) ] = Pr(-0.29 < Z < 0.29) = 2*Pr(Z<-0.29) = 0.77

5. 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.

6. 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”

7. 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.

8. 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.