PHP 2510 Inference about population mean; distribution of the sample mean; standard error; central limit theorem

Begin formalities of drawing inference from data Usual set up: target parameter (e.g. population mean) Use a statistic to estimate the parameter (e.g. sample mean) Recall: Statistic is a summary of data (random variables).

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Key Idea: We want to estimate population mean which is unknown. We know sample mean. (We collect data, and calculate sample mean.) We use sample mean as an estimate of the population mean. The sample mean is random. It has a distribution. We need to quantify uncertainty of using sample mean to estimate population mean.

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To motivate the idea...

Suppose a biostatistics program has 10 PhD students. We label them by A, B, ..., J. Their heights are (inch): 5.41, 5.31, 5.47, 5.54, 5.58, 5.84, 5.73, 5.41, 5.74, and 5.26. The population is: {A, B, ..., J}. Population average height (parameter of interest) is 5.529. Suppose that we do not know their average height, and want to estimate it using a random sample of 5 students. There are many ways to randomly choose 5 students.

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Possible samples include: sample

sample mean

1

BCDFH

5.514

2

BCEFH

5.522

3

DEHIJ

5.506

4

BDGIJ

5.516

5

ADEFI

5.622

...

...

...

We see only one of these, which is random.

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What if we decide to randomly select 8 students and use their average height as an estimate? Possible samples include: sample

sample mean

1

ABDEFHIJ

5.511

2

ABCDEFGH

5.536

3

ABCEGHIJ

5.489

4

BCDEFGIJ

5.559

5

ABCEFGHJ

5.501

...

...

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...

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5.8 5.7 5.6 5.5

x xxx xx xxx xx xx xx xx x

5.3

5.4

sample mean

x x x xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx x x

4

5

6

7

8

9

sample size

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Posing the basic question We want to draw inference about the mean of some population. We collect observations X1 , . . . , Xn and compute the sample mean X = (1/n)

n X

Xi

i=1

Note that each Xi is random. Question: What information about the true mean is captured by the sample mean? Specifically, what can we infer about the true mean from the sample mean?

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Sample mean is a random variable The key to conducting inference is to understand that the sample mean is a random variable. It has: • A mean • A variance • An associated probability model In short, we will study how the sample mean, which is observable, behaves in relation to the population mean, which is not.

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Intuition about the sample mean Much of what we will learn in a formal way appeals to intuition.

• A sample mean will almost never be equal to the population mean.

• A sample mean based on a large sample is expected to be closer to the population mean, compared to a sample mean based on a small sample.

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Example: Presidential polling The objective is to estimate percent of eligible voters who will vote for Barack Obama. The true percent who will vote for Obama is unknown. Let’s denote it by π. In the poll, the response of each individual is a Bernoulli random variable:   1 vote for Obama Xi =  0 do not vote for Obama We therefore can write both the mean and variance of Xi : E(Xi ) var(Xi )

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= π = π(1 − π)

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Three polls are taken, each using a different sample size: • 10 randomly chosen subjects

• 100 subjects

• 1000 subjects

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Example: Diastolic blood pressure Suppose we want to estimate mean DBP among those who are at risk for hypertension. The unknown mean is µ. The variance of DBP also is unknown; call it σ 2 . Random variable for the individual outcomes: Xi

= diastolic BP

• Mean of each outcome: E(Xi ) = µ • Variance of each outcome: var(Xi ) = σ 2

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Consider samples of size • 10 subjects

• 100 subjects

• 1000 subjects

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Mean and variance of sample mean Sample mean is a random variable: X

= (X1 + X2 + · · · + Xn )/n n X = Xi /n i=1

What is its mean and variance?

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Example 1: Presidential poll. What are the mean and variance of sample means?

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Example 2: Blood pressure Population distribution: Suppose DBP has normal distribution with µ = 80, σ = 10 What are the mean and variance of sample means?

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Distribution of the sample mean Case 1: Population distribution is normal For an individual in the population, Xi ∼ N (µ, σ 2 ) (Example: DBP has µ = 80, σ = 10) Then, for a sample of size n, the sample mean also has a normal distribution X n ∼ N (µ, σ 2 /n)

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Example: Population distribution of DBP is normal, with mean = 80, SD = 10 Sample 25 individuals, compute X. E(X) = var(X) = Moreover, X ∼

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Continuing with the DBP example, what is the distribution and variance of the following sample means? 1. Sample mean of n = 64 people

2. Sample mean of n = 100 people

3. Sample mean of n = 10000 people

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Distribution of the sample mean Case 2: Population distribution is not normal Examples: Poisson, Binomial Then, for large samples, the sample mean approximately has a normal distribution with mean equal to E(X) and variance equal to var(X)/n

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Example 1. Population distribution is Bernoulli with p = 0.2. What is the distribution of the following sample means? 1. Sample mean with n = 100 2. Sample mean with n = 64 3. What if p = 0.5 instead?

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Example 2. Population distribution is Poisson with λ = 1. What is the distribution of the following sample means? 1. Sample mean of 25 observations 2. Sample mean of 100 observations 3. What is the probability that a sample mean from n = 25 will exceed 1.2? What is the probability it will exceed 1.6?

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Standard error as a measure of precision The SD of a sample mean is called its standard error (SE). SE is a measure of how precisely the sample mean estimates the population mean. • Depends on n • Depends on underlying variation in the population being sampled For a sample drawn from a population having variance σ 2 , the √ standard error of X is σ / n • decreases with n • increases with σ

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Central Limit Theorem Characterizes distribution of X in large samples When n is large, X ∼ N (µ, σ 2 /n) where µ = σ2

=

population mean population variance

This is true for any underlying distribution of values. What is the implication?

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Summary so far Sample mean is a random variable The expected value of the sample mean is the same as the expected value for the population The variance of the sample mean is σ 2 /n √

The standard error of the sample mean is σ/ n When n is large, the sample mean follows a normal distribution If underlying distribution is normal, sample mean is normally distributed, regardless of sample size

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