Statistics for the Behavioral Sciences (5Statistics for the Behavioral Sciences (5 thth ed.) ed.) Gravetter & WallnauGravetter & Wallnau

Chapter 7Chapter 7

Probability and Samples:Probability and Samples:The Distribution of Sample MeansThe Distribution of Sample Means

University of GuelphPsychology 3320 — Dr. K. Hennig

Winter 2003 Term

Review: recall Table 4.1Review: recall Table 4.1

statement: (statement: (ss22 = = SSSS//nn-1)-1) proof: proof:

• collect the set of all possible samples for n=2 collect the set of all possible samples for n=2 selected from the populationselected from the population

• compute s2 for n and for n-1 and you’ll see for compute s2 for n and for n-1 and you’ll see for yourself: yourself: = 4 = the average of SS using = 4 = the average of SS using nn-1-1

Our proof begin with an analogy that Our proof begin with an analogy that has us knowing “reality” (population)has us knowing “reality” (population)

2

46

8

= “Reality” (population; “known unknown”)

e.g., p(King) given an old deck with some cards missing

= our sample

The distribution of sample meansThe distribution of sample means(defn)(defn) t the set of all possible he set of all possible randomrandom (w/ replacement) (w/ replacement) samples of size nsamples of size n

2

4

6

8

Fig.7-1

= ?? = ??

What is the probability of obtaining sample (2,4), or M>7?

ReminderReminder

We are not talking about the distribution We are not talking about the distribution of the sample but the distribution of of the sample but the distribution of means from the set of all possible samplesmeans from the set of all possible samples

Sample 1

Sample n...

X X 1 2 3 4 5

The distribution of sample means for n=2The distribution of sample means for n=2

But we can’t draw the complete set of samples, thus…But we can’t draw the complete set of samples, thus… Central Limit Theorem: M = Central Limit Theorem: M = and standard deviation = and standard deviation = //n as n as nn - -

> infinity> infinity Fig. 7.3

Distribution of sample means (contd.)Distribution of sample means (contd.)

The distribution of sample means will be The distribution of sample means will be normal if … pick one:normal if … pick one:• the population sample is normally distributed, orthe population sample is normally distributed, or• n is relatively large, i.e., >30n is relatively large, i.e., >30

In most situations with n > 30 the distribution In most situations with n > 30 the distribution of means will be normal regardless of the of means will be normal regardless of the distribution of the population. Note. M (of distribution of the population. Note. M (of sample means) is an sample means) is an unbiasedunbiased stat; cf. M in stat; cf. M in chapter 4 - had to correct schapter 4 - had to correct s22

The expected valueThe expected value

Distribution of sample means (contd.)Distribution of sample means (contd.)

expected value of Mexpected value of M will equal will equal Thus far to describe samples we’ve used:Thus far to describe samples we’ve used:

• central tendency (central tendency (meanmean, median, mode), median, mode)• shape (or variability of the sample)shape (or variability of the sample)

Similarly of the distribution of sample meansSimilarly of the distribution of sample means• meanmean• now add, shape or standard deviation, called now add, shape or standard deviation, called standard error standard error

of Mof M (i.e., a sample will not perfectly represent the (i.e., a sample will not perfectly represent the population)population)

= = MM = standard distance between M and = standard distance between M and

Magnitude of the error is related to: size of sample Magnitude of the error is related to: size of sample and standard dev of the populationand standard dev of the population

Distribution of sample means (contd.)Distribution of sample means (contd.)

The law of large numbersThe law of large numbers• M = M = • standard error of M decreasesstandard error of M decreases

Formula for standard error:Formula for standard error:

nM

More about standard errorMore about standard error

standard error vs. standard deviation?????standard error vs. standard deviation?????• whenever you are talking about a whenever you are talking about a samplesample use use

standard error, i.e., alwaysstandard error, i.e., always• always some error: 50% of Ms < always some error: 50% of Ms <

When n = 1, then When n = 1, then MM== (recall: for the full (recall: for the full set of samples n = 1)set of samples n = 1)

Looking ahead to inferential statisticsLooking ahead to inferential statistics

rat pups are treated with a growth rat pups are treated with a growth hormone: hormone: =400 but not all same size =400 but not all same size =20, treat a sample of 25=20, treat a sample of 25

nM

M

Mz

Standard error as an estimate of Standard error as an estimate of reliabilityreliability The problem:The problem:

• I must estimate the population from a sampleI must estimate the population from a sample• but if I had a different sample, I would obtain a but if I had a different sample, I would obtain a

different resultsdifferent results• the question becomes: would the first sample be the question becomes: would the first sample be

similar to the second sample (or 3similar to the second sample (or 3rdrd or 4 or 4thth, etc.), etc.)• in the previous example it was easy to find two in the previous example it was easy to find two

samples with very different meanssamples with very different means• by increasing n, you increase your confidence by increasing n, you increase your confidence

that your sample is a measure of your populationthat your sample is a measure of your population

Summary (p. 222)Summary (p. 222)

1)1) What is the distribution of sample means? What is the distribution of sample means? the set of all Ms for all possible random the set of all Ms for all possible random samples for sample size n for a given samples for sample size n for a given population.population.1)1) shape: population must be N or n>30shape: population must be N or n>30

2)2) central tendency: M = central tendency: M = 3)3) variability: variability:

2)2) Standard error: the standard deviation of (1) Standard error: the standard deviation of (1) - tells us how much error to expect if using a - tells us how much error to expect if using a sample to estimate a populationsample to estimate a population

3)3) Location of M in the distribution Location of M in the distribution

nM

Learning check (p. 209 Q#1)Learning check (p. 209 Q#1)

Population of scores is normal Population of scores is normal =80 =80 =20=20• describe the distribution of sample means of describe the distribution of sample means of

size n = 16: shape? central tendency? size n = 16: shape? central tendency? variability?variability?

• What if n = 100?What if n = 100?