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
1 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
2 THE DISTRIBUTION OF SAMPLE MEANS
Population of 4 scores: 2 4 6 8 = 5
4 random samples (n = 2):
X 1= 4 X 3 = 5
X 2 = 6 X 4 = 3
X is rarely exactly
Most X a little bigger or smaller than
Most X will cluster around
Extreme low or high values of X are relatively rare
With larger n, X s will cluster closer to µ (the DSM will have smaller error, smaller variance)
3 A Distribution of Sample Means
X = 4 X = 5 X = 6
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.
4 THE DISTRIBUTION OF SAMPLE MEANS
A distribution of sample means (X ) All possible random samples of size n A distribution of a statistic (not raw scores)
“Sampling Distribution” of X
Probability of getting an X , given known and Important properties
(1) Mean (2) Standard Deviation (3) Shape
5 PROPERTIES OF THE DSM
Mean?
X =
Called expected value of X
X is an unbiased estimate of Standard Deviation?
Any X can be viewed as a deviation from
X = Standard Error of the Mean
X = n
Variability of X around
Special type of standard deviation, type of “error”
Average amount by which X deviates from
6 Less error = better, more reliable, estimate of population parameter
X influenced by two things:
(1) Sample size (n)
Larger n = smaller standard errors
Note: when n = 1 X =
as “starting point” for X,
X gets smaller as n increases
(2) Variability in population ()
Larger = larger standard errors
Note: X = M
7 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 σ = 20. Notice that the size of the standard error decreases as the sample size increases.
8 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
9 Why are Sampling Distributions important?
Tells us probability of getting X , 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
10 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 = X x
Example #1: = 80 = 12
What is probability of getting X 86 if n = 9?
11 Example #1b: = 80 = 12
What if we change n =36
What is probability of getting X 86
12 Example #2:
= 80 = 12
What X marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9)
13 Homework problems: Chapter 7: 3, 10, 11, 17
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