STAT 113 Inference Using Normal Approximations Colin Reimer Dawson - PowerPoint PPT Presentation

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal STAT 113 Inference Using Normal Approximations Colin Reimer Dawson Oberlin College November 4, 2020 1 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal 2 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Outline Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal 3 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal P -value = Proportion of Randomized Sample Statistics Number of Simulated Datasets 1500 AtLeast270Heads 1000 FALSE TRUE 500 0 0.45 0.50 0.55 0.60 Proportion of Heads in 500 Flips Figure: Randomization distribution for the number of heads in 500 coin flips, highlighting the one-tailed P -value testing H 1 : p > 0 . 5 for an observation of 270 heads. 4 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Confidence Level ≈ Proportion of Bootstrap Samples Number of Bootstrap Datasets 2000 1500 InInterval FALSE 1000 TRUE 500 0 0.3 0.4 0.5 0.6 0.7 Mean Mercury Level in 50 Observations (parts per million) Figure: Bootstrap distribution for mean mercury level in fish in Florida Lakes (from FloridaLakes dataset). The middle 95% is highlighted illustrating a 95% confidence interval. 5 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? • A: The Central Limit Theorem 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? • A: The Central Limit Theorem • One of the most important results in probability 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? • A: The Central Limit Theorem • One of the most important results in probability For sufficiently large datasets, sampling distributions of sMeans have approximately a Normal (bell-shaped) distribution . 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? • A: The Central Limit Theorem • One of the most important results in probability For sufficiently large datasets, sampling distributions of sMeans have approximately a Normal (bell-shaped) distribution . 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal So what’s with all these bell shapes? • Q: Why are so many distributions “bell-shaped”? • A: The Central Limit Theorem • One of the most important results in probability For sufficiently large datasets, sampling distributions of sMeans have approximately a Normal (bell-shaped) distribution . 6 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Corollaries of the Central Limit Theorem This theorem, together with some other properties of Normal distributions, implies that, for large enough datasets: • Sampling distributions of means are approximately Normal 7 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Corollaries of the Central Limit Theorem This theorem, together with some other properties of Normal distributions, implies that, for large enough datasets: • Sampling distributions of means are approximately Normal • Sampling distributions of proportions are approximately Normal 7 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Corollaries of the Central Limit Theorem This theorem, together with some other properties of Normal distributions, implies that, for large enough datasets: • Sampling distributions of means are approximately Normal • Sampling distributions of proportions are approximately Normal • Sampling distributions of differences of means are approximately Normal 7 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Corollaries of the Central Limit Theorem This theorem, together with some other properties of Normal distributions, implies that, for large enough datasets: • Sampling distributions of means are approximately Normal • Sampling distributions of proportions are approximately Normal • Sampling distributions of differences of means are approximately Normal • Sampling distributions of differences of proportions is approximately Normal 7 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Corollaries of the Central Limit Theorem This theorem, together with some other properties of Normal distributions, implies that, for large enough datasets: • Sampling distributions of means are approximately Normal • Sampling distributions of proportions are approximately Normal • Sampling distributions of differences of means are approximately Normal • Sampling distributions of differences of proportions is approximately Normal • Sampling distributions of regression slopes are approximately Normal (when regression conditions are met) 7 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Density Functions 0.025 0.020 0.015 Density 0.010 0.005 0.000 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 Birth Weight (oz) Figure: Densities of Babies’ Birth Weights (Nolan and Speed, 2000) 8 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Proportion = Area Under the Density Curve 0.025 0.02 0.015 Density 0.01 Shaded = 0.06 of total 0.005 0 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 Birthweight in oz Figure: Approximating birth weight distribution using a Normal. Shaded area is the proportion of the distribution at or above 148 oz 9 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Outline Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal 10 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Normal Distributions Normal distributions are completely specified by their mean ( µ ) and their standard deviation ( σ ). We can write N (0 , 1) as shorthand for a Normal with mean 0 and standard deviation 1. 1.5 N(0, 1) N(2, 1) N(0, 0.5) 1.0 N(−4, 0.3) Density 0.5 0.0 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Some Variable 11 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Area Under Normal Curve 0.4 0.3 Density 0.2 0.1 0.0 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Some Variable Can we work out these areas without simulation? 12 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal StatKey to the Rescue! 13 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal R Works Too library(mosaic) ## Area under the curve to the right of 1.5 ## First argument: the cutoff value ## mean, sd: the mean and standard deviation of the Normal ## lower.tail = TRUE/FALSE; area to the left (TRUE) or right (FALSE) xpnorm(1.5, mean = 0, sd = 1, lower.tail = FALSE) z = 1.5 0.4 0.3 density 0.2 0.1 0.0 −4 −2 0 2 4 x [1] 0.0668072 ## Creates a plot showing the z-score of the cutoff ## and returns the proportion under the curve on the specified side 14 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Self-Check Find the specified tail proportions for a Normal distribution (use either StatKey or R, or use both to confirm your results!) 1. The proportion of cases above 62 in a N(50,10) distribution 2. The proportion of cases below 8 in a N(10,2) distribution. 15 / 33

Analytic Approximations Normal Distributions Hypothesis Tests Using a Normal Confidence Intervals Using a Normal Finding the Cutoff if We Know the Tail Proportion 16 / 33