THE SAMPLING DISTRIBUTION OF THE MEAN



The Distribution of Sample Means

Inferential statistics:

Generalize from a sample to a population

Statistics vs. Parameters

Why?

Population not often possible

Limitation:

Sample won’t precisely reflect population

Samples from same population vary

“sampling variability”

Sampling error = discrepancy between sample statistic and population parameter

• Extend z-scores and normal curve to SAMPLE MEANS rather than individual scores

• How well will a sample describe a population?

• What is probability of selecting a sample that has a certain mean?

• Sample size will be critical

• Larger samples are more representative

• Larger samples = smaller error

The Distribution of Sample Means

Population of 4 scores: 2 4 6 8 ( ( = 5

4 random samples (n = 2):

[pic]1= 4 [pic]3 = 5

[pic]2 = 6 [pic]4 = 3

[pic] is rarely exactly (

Most [pic] a little bigger or smaller than (

Most [pic] will cluster around (

Extreme low or high values of [pic] are relatively rare

With larger n, [pic]s will cluster closer to µ (the DSM will have smaller error, smaller variance)

A Distribution of Sample Means

The Distribution of Sample Means

1. A distribution of sample means ([pic])

2. All possible random samples of size n

3. A distribution of a statistic (not raw scores)

“Sampling Distribution” of [pic]

4. Probability of getting an [pic], given known ( and (

5. Important properties

(1) Mean

(2) Standard Deviation

(3) Shape

Properties of the DSM

Mean?

([pic] = (

Called expected value of [pic]

[pic] is an unbiased estimate of (

Standard Deviation?

Any [pic] can be viewed as a deviation from (

([pic] = Standard Error of the Mean

([pic] = [pic]

Variability of [pic] around (

Special type of standard deviation, type of “error”

Average amount by which [pic] deviates from (

Less error = better, more reliable, estimate of

population parameter

([pic] influenced by two things:

(1) Sample size (n)

Larger n = smaller standard errors

Note: when n = 1 ( ([pic] = (

( as “starting point” for ([pic],

([pic] gets smaller as n increases

(2) Variability in population (()

Larger ( = larger standard errors

Note: ([pic] = (M

Shape of the DSM?

Central Limit = DSM will approach a normal dist’n

Theorem as n approaches infinity

Very important!

True even when raw scores NOT normal!

True regardless of ( or (

What about sample size?

(1) If raw scores ARE normal, any n will do

(2) If raw scores NOT normal, n must be

“sufficiently large”

For most distributions ( n ( 30

Why are Sampling Distributions important?

• Tells us probability of getting [pic], given ( & (

• Distribution of a STATISTIC rather than raw scores

• Theoretical probability distribution

• Critical for inferential statistics!

• Allows us to estimate likelihood of making an error when generalizing from sample to popl’n

• Standard error = variability due to chance

• Allows us to estimate population parameters

• Allows us to compare differences between sample means – due to chance or to experimental treatment?

• Sampling distribution is the most fundamental concept underlying all statistical tests

Working with the

Distribution of Sample Means

If we assume DSM is normal

▪ If we know ( & (

▪ We can use Normal Curve & Unit Normal Table!

z =

Example #1:

( = 80 ( = 12

What is probability of getting [pic] ( 86 if n = 9?

Example #1b:

( = 80 ( = 12

What if we change n =36

What is probability of getting [pic] ( 86

Example #2:

( = 80 ( = 12

What [pic] marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9)

Homework problems:

Chapter 7: 3, 10, 11, 17

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Figure 7-7 (p.215)

The distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with µ = 80 and à = 20. Notice that the si = 100 obtained from a normal population with µ = 80 and σ = 20. Notice that the size of the standard error decreases as the sample size increases.

Figure 7-3 (p. 205)

The distribution of sample means for n = 2. This distribution shows the 16 sample means obtained by taking all possible random samples of size n=2 that can be drawn from the population of 4 scores (see Table 7.1 in text). The known population mean from which these samples were drawn is µ = 5.

[pic]= 4

[pic]= 5

[pic]= 6

[pic]

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