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Analytic Approximations Density Functions Properties of Normal Distributions Normal Bootstrap Distibution STAT 113 Working with Theoretical Distributions Colin Reimer Dawson Oberlin College November 2, 2017 1 / 26 Analytic Approximations

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Analytic Approximations Density Functions Properties of Normal Distributions Normal Bootstrap Distibution

STAT 113 Working with Theoretical Distributions

Colin Reimer Dawson

Oberlin College

November 2, 2017 1 / 26

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Outline

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P-value = Proportion of Randomized Sample Statistics

200 220 240 260 280 300 0.00 0.02 0.04 Values Probability

  • ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
  • ● ● ● ●
  • ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

P(X ≥ 270) ≈ 0.04

Figure: Randomization distribution for the number of heads in 500 coin flips, highlighting the one-tailed P-value testing H1 : p > 0.5 for an

  • bservation of 270 heads.

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Confidence Level = Proportion of Bootstrap Samples

xMercury Density

2 4 6 8 0.4 0.5 0.6 0.7

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.

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Properties of Sampling Distributions

Most (about 95%) of simple random samples have a sample mean (¯ x) which is within 2 Standard Errors of the population mean (µ). Therefore, about 95% of the time, the population mean will be within 2SE of the sample mean! A similar statement holds for some other statistics/parameters, under a particular condition. What condition? The sampling distribution needs to be (approximately) symmetric and bell-shaped 6 / 26

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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 samples, sample means have a Normal (bell-shaped) distribution. 7 / 26

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Sample Means Show Up A Lot

  • Sample means are sample means (did you know this?)
  • Sample proportions are sample means (encode binary variable

as 0s and 1s) 8 / 26

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Even More Stuff is Normal

Also...

  • Sum of two Normals is Normal
  • Rescaling a Normal by a constant is Normal
  • Difference of Normals is Normal

So...

  • Sampling distribution for difference of sample means is

approximately Normal

  • Sampling distribution for difference of sample proportions is

approximately Normal 9 / 26

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Outline

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Approximating with a Smooth Curve

Birthweight in oz Frequency 50 100 150 200 400 800 Birthweight in oz Frequency 50 100 150 200 200 400 Birthweight in oz Frequency 50 100 150 200 100 200 Birthweight in oz Frequency 50 100 150 200 100 200 Birthweight in oz Frequency 50 100 150 200 50 100 150 Birthweight in oz Frequency 50 100 150 200 20 40 60

Figure: Frequency Histograms of Babies’ Birth Weights (Nolan and Speed, 2000)

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Density

Proportion = Area = Height × Width Density = Height = Proportion Width This quantity (proportion divided by width) is called “density” by analogy to physics: “amount of stuff” divided by “amount of space”. 12 / 26

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Density Histograms

Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030

Figure: Density Histograms of Babies’ Birth Weights (Nolan and Speed, 2000)

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Density Functions

Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030 Birthweight in oz Density 50 100 150 200 0.000 0.015 0.030

Figure: Densities of Babies’ Birth Weights (Nolan and Speed, 2000)

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Proportion = Area Under the Density Curve

50 100 150 200 0.000 0.010 0.020 Birthweight in oz Density P = 0.067

Figure: Approximating birth weight distribution using a Normal. Shaded area

is P(Weight ≥ 148 oz)

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Outline

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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.

−6 −4 −2 2 4 6 0.0 0.5 1.0 1.5 x Density N(0, 1) N(2, 1) N(0, 0.5) N(−4, 0.3)

density(x) = 1 σ √ 2πe−( x−µ

σ ) 2

, but we won’t use this directly. 17 / 26

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Normal Distributions

µ − 2σ µ − σ µ µ + σ µ + 2σ

Pairs: (Approximately) what proportion of the area under the curve is shaded? In a bell-shaped (normal) distribution, 95% of cases lie within 2 standard deviations of the mean. So 5% lie beyond 2σ from µ. 18 / 26

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Area Under Normal Curve

−2 −1 1 2

Area under a curve using calculus: ∞

1.5

1 σ √ 2πe−( x−0

1 ) 2

dx but this integrand doesn’t have a closed-form antiderivative 19 / 26

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StatKey to the Rescue!

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R Works Too

library("mosaic") ## Area to the right of 1.5 xpnorm(1.5, mean = 0, sd = 1, lower.tail = FALSE) If X ~ N(0, 1), then P(X <= 1.5) = P(Z <= 1.5) = 0.9331928 P(X > 1.5) = P(Z > 1.5) = 0.0668072

density

0.1 0.2 0.3 0.4 0.5 −2 2

1.5

(z=1.5) 0.9332 0.0668

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Outline

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Quantiles of a Normal Curve

Suppose that the bootstrap distribution of means for samples of size 500 Atlanta commute times is N(29.11, 0.93). Find an endpoint (percentile) so that just 5% of the bootstrap means are smaller. 23 / 26

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StatKey...

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And in R ...

xqnorm(0.05, mean = 29.11, sd = 0.93) P(X <= 27.5802861269351) = 0.05 P(X > 27.5802861269351) = 0.95

density

0.1 0.2 0.3 0.4 0.5 26 28 30 32

27.5803 (z=−1.645) 0.05 0.95

[1] 27.58029

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Goals

Confidence Intervals

If we can approximate a bootstrap distribution with a Normal, we can construct a confidence interval.

P-values

If we can approximate a randomization distribution with a Normal, we can compute P-values. 26 / 26