Stats chapter 9

Chapter 9

Sampling Distributions

9.1 SAMPLING DISTRIBUTIONS

Definitions

Parameter• the value of a characteristic for the

entire population attained through census

• in practice, is usually an unknown or estimated value

Definitions

Statistic• the value of a characteristic for the

entire population attained through sampling

• In practice, the value of a statistic is used to estimate the parameter

Sampling Variability

• Random samples will produce different values for a statistic

• The statistics are usually not the same value of the parameter

• Different sample produce different values (all of which are “close” to the parameter)

• This fact is known as sampling variability

• The value of a statistic for the same parameter varies in repeated sampling.

Parameters Statistics

• Parameter Statistic

• Mean of a Pop Mean of a sample

• Prop. of a pop. Prop. of a sample

x

p p

Sampling Distribution

• All samples of size n are taken from a population of size N

• A histogram of these sample statistics is created

• This distribution is called the “sampling distribution”

• In practice, the sampling distribution is theorized, but never “created”

Creating a Sampling Distribution

• Let’s look at a pop N = 5, who answered ‘yes’ or ‘no’ to the question “Do you like toast?”

• We want to know proportion who say ‘yes’• Here are the responses:• ID Response

01 Yes02 No03 Yes04 No05 Yes

Creating a Sampling Distribution

• Let’s look at each sample and the phat for sample size n = 3

• Sample # ID’s in sample p-hat• 1 01, 02, 03 0.66

2 01, 02, 04 0.333 01, 02, 05 0.664 01, 03, 04 0.665 01, 03, 05 16 01, 04, 05 0.667 02, 03, 04 0.338 02, 03, 05 0.669 02, 04, 05 0.3310 03, 04, 05 0.66

• You can imagine that this quickly gets labor intensive!

Creating a Sampling Distribution

• Create a Histogram• Class Count

0.00-0.24 00.25-0.49 30.50-0.74 60.75-1.00 1

• Notice that p = 0.6, and the mean of this distribution is approx 0.6

0 0.5 1

7654321

Describing Sampling Distributions

• Like most 1-var data, we describe :– Center– Shape– Spread– Unusual features/Outliers

• If you are using a sample to estimate a parameter, of the sampling distribution:– Where should the center be?– What about the “ideal shape?”– What would you like the spread to be?– Would outliers be helpful?

Sampling Distribution and Bias

• When a statistic is unbiased, the mean of the sampling distribution is the value of the parameter.– This is actually a pretty powerful statement. – In order to find the value of the parameter, you just

need to take a lot of samples! (wait, that’s not good either)

– Revision: If a statistic is unbiased, then “chances are” the value of any sample should be close to the value of parameter

• Statistics that are unbiased are called “unbiased estimators” (these are good)

Variability of a Statistic

• The spread of a sampling distribution is known as the variability of the statistic

• Large sample size = less variability

The Enemies of Sampling

• Enemy #1: Bias• Enemy #2: Variability• A visual of the difference:

The Enemies of Sampling

• Another look with Histograms:

9.2 SAMPLE PROPORTIONS

Sampling Distribution for Proportions

• For each sample, calculate p-hat:

• The sampling distribution of p-hat will have:–Mean = p (the parameter)– Standard deviation:

# of successes

sample size

Xp

n

p

pq

n

Sampling Distribution for Proportions

• Notice that this is an unbiased estimator!

• The standard deviation decreases when the sample size is large

• Std. Dev. and sample size have an “inverse square” relation– Ex. If we want ½ the std dev,

we need to 4x the sample size– Ex. If we want to 1/3 the std dev,

we need to 9x the sample size

Sampling Distribution for Proportions

• We will (almost) always use the Normal approximation for the sampling distribution for p-hat.

• This means we will need some conditions:1. We want “N > 10n”

This ensures our std dev formula holds2. np > 10 and nq > 10

This ensures our samp. dist. is approx. Normal

Samp Dist for Prop. (Example)

We are sampling from a large population. Our sample size is 1500. We know that the p = 0.35. What is the probability that our sample is more than 2 percent from the parameter?

Samp Dist for Prop. (Example)

• To summarize the problem, we are trying to find out what proportion of samples have a p-hat greater than 0.37 or less than 0.33

• It will be easier to use the rules of compliments and to find “1 – P(0.33 < p-hat < 0.37)”

Samp Dist for Prop. (Example)

• Can we use a Normal approximation for this problem? Let’s check the conditions:1. Although we are not told the exact

population size N, we are told the population is large.

“We are told the population is large, so N > 10(1500)”

Tip: when a problem says the population is large, you are to interpret that the population is greater than 10n

Samp Dist for Prop. (Example)

• Can we use a Normal approximation for this problem? Let’s check the conditions:2. np = 1500(0.35) = 525 > 10

nq = 1500(0.65) = 975 > 10

“Since np = 525 > 10 and nq = 975 >10 and N > 10(1500), we can use the Normal distribution”

• Note: It is extremely important that you state and justify the use of the Normal distribution.

Samp Dist for Prop. (Example)

• Time for a graph (before normalization)Remember, you don’t have to be too fancy here!

Samp Dist for Prop. (Example)

• Let’s Normalize!

0.35

0.35 0.650.0123

1500

p

pq

n

0.33 or 0.37 1 0.33 0.37P p p P p

0.33 0.35 0.37 0.351

0.123 0.123P z

1 1.63 1.63P z

Samp Dist for Prop. (Example)

• Now the normalized graph

Samp Dist for Prop. (Example)

• Compute the area 0.33 or 0.37 1 0.33 0.37P p p P p

0.33 0.35 0.37 0.351

0.123 0.123P z

1 1.63 1.63P z

use "Normcdf(-1.63, 1.63)"1 0.8968

0.1032

Samp Dist for Prop. (Example)

• Finish the normalized graph

Samp Dist for Prop. (Example)

• Summary:– “The probability that a sample (n=1500) is

more than 2 percent from the parameter is 0.1032”

• Notes: remember that in this context, probability is the same as proportion, and proportion is the same as area.

• Actually, you’ve done many of these kinds of problems already, right?

9.3 SAMPLE MEANS

Samples vs. Census

• Histogram for returns on common stocks in 1987:

• Histogram for 5 stock portfolios in 1987

Samples vs. Census

• We can see from the previous slide that the distribution of samples (portfolio)– Are less variable than the census– Are more Normal than the census

Sampling Distribution for Means

• Suppose we have a sampling distribution of samples size n from a large population

• The mean of the sampling distribution is the mean of the population

• The std dev of the samp dist is given by:

x

x n

Sampling Distribution for Means

• The sample mean is an unbiased estimator of the population mean

• Like for proportions, the std dev and the population size have an inverse square relation

• Like for proportions, we need N > 10n for our std dev formula to hold up

• This sampling distribution holds true even if the population is not Normal!

The Central Limit Theorem

• An SRS of size n from any population will produce a sampling distribution that is N( , /(n)) whenever n is large enough.

• Caution: this theorem is only true for means. Do not try to use the CLT for proportions!

The Central Limit Theorem

Why we use CLT:• From the previous section, we saw

that we use the Normal dist to gauge probability of producing samples

• We invoke the CLT to justify usage of the Normal distribution– Using Normal dist w/o justification is a

“nono”

The Central Limit Theorem

When to use the CLT:• Sampling Distribution for a mean ()• We need to Normalize the sample

mean• The sample is described as “large”– Generally, n > 30

• The raw data is not given