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```Week 2
Sampling distributions and
testing hypotheses
handout available at
http://homepages.gold.ac.uk/aphome
Trevor Thompson
8-10-2007
1
Review of following topics:

1) How individual scores are distributed

2) How mean scores are distributed

3) One-sample z-test

testing the difference between a sample mean
and a known population mean
- Howell (2002) Chap 4 & 7. ‘Statistical Methods for Psychology’
2
Distribution of individual scores

Uniform distribution
Dice scores are uniformly
distributed (each score has
an equal probability of
occurrence)

Normal distribution
Scores on many variables
are normally
distributed (e.g. IQ)

Sampling distribution is not always identical to population
distribution because of ‘sampling error’
sample of 600 die throws
How individual values are distributed depends on the
nature of these values
sample of 5000 IQ scores

3
Distribution of individual scores



What is probability (p-value) of
randomly sampling one person with an
≈ 50%
IQ of 100 or more?
What is probability of randomly selecting
an IQ score of 130+? (We can calculate
this from what we know about the
properties of the normal curve) ≈ 2.5%
The probability for any IQ can be
calculated* – calculate the z-score (i.e.
the number of SD’s above or below the
mean), then look up corresponding pvalue in table (or use SPSS CDF function)
(*assuming population parameters of M=100, SD=15 and normal distribution)
4

1) How individual scores are distributed

2) How mean scores are distributed

3) One-sample z-test
5
Sampling distribution of means
Q: What is the probability of a group of 36 having a mean IQ of 106 ?


We need to know how means are
Specifically, we need to know (as with
individual scores):



1) Mean same as mean of individual scores (100)
2) Shape of distribution next slide
3) Standard deviation
6
Shape of distribution of means
I repeatedly sampled 36 scores and calculated the mean. I then repeated this
several thousand times and plotted these means:
Sample 1: Random sample of 36 scores produced M=103.5
Sample 2: Random sample of 36 scores produced M=100
Sample 3: Random sample of 36 scores produced M=96
Sample 4: Random sample of 36 scores produced M=102
Sample 5: Random sample of 36 scores produced M=100
95 97.5 100 102.5 105
The distribution
of means appears
to be the same as
individual scores!
– i.e. normal
7
Sampling distribution of means




1) Mean – equal to the population mean (100)
2) Shape - normal
3) Standard deviation – how widely are the
Mean scores are spread more closely around
the centre than individual scores

this makes intuitive sense – while individual scores
of 130 are not exceptionally rare (p=2.5%), mean
IQs of 130 would be extremely rare when group
size is 1,000!
8
Sampling distribution of means




In fact, we can calculate precisely the spread of mean
scores around the centre:
SEM= Sx
√N
SEM (the standard error of the mean) is the standard
deviation of the mean (rather than standard deviation of
individual scores)
The above formula shows that the bigger the sample
size (N), the smaller the SEM – i.e. the more closely
scores are clustered around the population mean
9
Sampling distribution of means
Q: What is the probability of N=36 having a mean IQ of at least 106 ?

We can now plot the sampling distribution of the
means. We know the shape is normal, M=100 &
SEM =15 = 2.5
√36
So, if we know how
individual scores are
distributed (i.e.
shape, M & SD) we
also know how means
are distributed and
can test hypotheses
10
Confidence Limits



The 95% confidence limits is the range within which
95% of the sample means will fall
If a sample mean lies within these limits then we
cannot reject the null hypothesis
Our value of 106 lies outside these limits –we reject
the null hypothesis (p<.05!)
11

1) How individual scores are distributed

2) How mean scores are distributed

3) One-sample z-test
12
One sample z-test


One sample z-test: Compares the mean score of one
group against a population mean. This can only be
performed when we know the population mean and the
population standard deviation
To perform a one-sample z-test, calculate how many
SEMs above/below the population mean your sample
mean is. Expressed as a formula:
z=
X–μ
(where SEM=σ/√N)
SEM

A one-sample z-test is what we have previously performed!
z= (106 – 100)/2.5
z=2.4, which gives p<.05 –significant!
13
One sample t-test




A one sample z-test is used when we already know the
population mean and SD
A one sample t-test is used when we know the
population mean (μ) but not the population SD (σ)
As we do not know σ, we estimate it from s (sample SD).
But, as s is often too small, the p-value is inaccurate when
using z-distribution tables. Use t-distribution tables for
more conservative p-values
To perform a one-sample t-test:
t
=X–μ
(where SEM=s/√N)
SEM
but look up p-value from a t not a z-distribution table
14
Central Limit Theorem


Everything we have done so far is explained by central
limit theorem
‘Given a population with mean, , and standard
deviation, , the sampling distribution of the mean will
have’:
(i) a mean equal to 
(ii) a standard deviation equal to /√N, where N is
the sample size
(iii) a distribution which will approach the normal
distribution as N increases
15
Central Limit Theorem

The approximation of the sampling distribution of the
mean to a normal distribution is true -whatever the shape
of the distribution of individual values
distribution of single die scores
distribution of mean dice scores
Populati on
Samp. Dist. of Mean
Normal
Symmetric and unimodal
Highly skewed
Normal for all N
Normal for s mall N
Normal for N>30
16
One sample z-tests - examples
1) The mean IQ of a group of 16 people was measured as
103. Is this significantly different from the population
using the population parameters previous specified?



No, SEM=3.75 (15/√16)
z=0.8 (103-100/3.75)
p>0.5 – non-significant
2) Is a sample of 25 dice throws, with a mean score of
3.85, sampled from a fair die? [=3.5, =1.7]
No, SEM=0.34 (1.7/√25)
z≈1 (3.85-3.5/0.34)
p>0.5 – non-significant
17
Summary

How to perform a one sample z-test

How to perform a one sample t-test

Underlying logic behind one sample tests

Rules of central limit theorem
18
```
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