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• From last lecture (Sampling Distribution):
– The first important bit we need to know about
sampling distribution is…?
– What is the mean of the sampling distribution
of means?
– How do we calculate the standard deviation
of the sampling distribution of means and
what is it called?
Hypothesis testing:
The story so far...
(a) The means of a sampling distribution are often
normally distributed (even if the original population
from which the samples came is not normal).
This is true of N ≥ 30.
(b) Any given sample mean ( X ) can be expressed
in terms of how much it differs from the mean of
the sampling distribution (i.e., as a z-score).

Brain size in hares: sample means and population means
population: mean = 500 g
sample A: mean = 650g
sample C: mean = 500g
sample B: mean = 450g
sample D: mean = 600g
The Central Limit Theorem in action:
Frequency with which each sample mean
occurs:
sample A mean
sample F mean
sample G mean
sample B mean
sample H mean
sample C mean
sample J mean
sample D mean
sample E mean
sample K mean,
etc.....
the population
mean, and the
mean of the
sample means
(c) Any given sample mean can be expressed in terms
of how much it differs from the population mean.
Population mean
A particular sample mean
(d) "Deviation from the mean" is the same as "probability
of occurrence": a sample mean which is very deviant
from the population mean is unlikely to occur.
A word about the use of the term
“population”
• The term does not necessarily refer to a set
of individuals or items (e.g. cars). Rather, it
refers to a state of individuals or items.
• Example: After a major earthquake in a city
(in which no one died) the actual set of
individuals remains the same. But the anxiety
level, for example, may change. The anxiety
level of the individuals before and after the
quake defines them as two populations.
A concrete example:
An American Sludge Oil Company has
developed a new petrol additive that they
claim will increase the mileage of cars
• It is known that the current average of
cars is 18 mi/gallon
• A sample of 100 cars is taken and run
on sludge fuel
• The sample mileage is 19 mi/gallon
• Is the Sludge company’s claim valid?
• When the test is fair ONLY two sources
contribute to the difference between the
population and the sample:
– The effect of the fuel additive
– Random variability
Our job is to decide which of these possibilities is it
Hypothesis testing
• Two types hypotheses: H0 and H1
• The NULL hypothesis (H0): The mean of
the population from which the sample
comes IS 18 mi/gallon.
• The ALTERNATIVE hypothesis (H1):
The mean of the population from which
the sample comes IS SIGNIFICANTLY
DIFFERENT FROM 18 mi/gallon.
The question we want to ask is: How
likely is it that the sample (N = 100)
whose mean is 19 mi/gallon comes
from a population whose mean and
standard deviation are18 mi/gallon
and 4 mi/gallon, respectively?
Assume we know the following:
• The population mean mileage of cars:
k = 18 mi/gallon
• The population standard deviation (σ) = 4
mi/gallon
• The sample mean (X ) = 19 mi/gallon
• Number of cars (N) = 100

The sampling distribution for N = 100
Frequency
of sample
means
17
µ = 18

19
4
X 

 0.4
N
100
We start by supposing the null hypothesis is true.
Meaning that the sampling distribution from which
our sample comes is one for which µ = 18
Then we ask what is the probability that our
sample came from this distribution. If the
probability is low, then we reject the null
hypothesis in favor of the alternative hypothesis.
Assume "innocent until proven guilty": we retain the null
hypothesis unless there is enough evidence to allow us to
reject it in favour of the alternative hypothesis.
Large difference between sample mean and µ: reject H0,
and assume H1 is true.
If the difference is Small: have no reason to reject H0.
How low does the probability have to be?
• We usually select 0.05 as the cutoff. (Sometimes we also
select 0.01). This is called the level of significance.
• We look for X values that occur with a probability of 0.05
or less.

Frequency
of sample
means
17
µ = 18
19
How low does the probability have to be?
• We usually select 0.05 as the cutoff. (Sometimes we also
select 0.01). This is called the level of significance.
• We look for X values that occur with a probability of 0.05
or less.

Frequency
of sample
means
17
µ = 18
19
How low does the probability have to be?
• We usually select 0.05 as the cutoff. (Sometimes we also
select 0.01). This is called the level of significance.
• We look for X values that occur with a probability of 0.05
or less.

Frequency
of sample
means
17
µ = 18
19
The shaded area corresponds to the 5% of the lowest
and highest means (X ) in the sampling distribution

Frequency
of sample
means
0.025
17
0.025
µ = 18
19
Next we find the critical values
These are given as z-scores in the Table:
Areas Under the Unit Normal Distribution
Frequency
of sample
means
upper
critical value
1.96
lower
critical value
-1.96
0.025
17
0.025
µ = 18
19
A portion of the table:
z
Below z
Above z
1.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
0.9713
0.9719
0.9725
0.9732
0.9738
0.9744
0.9750
0.9755
0.9761
0.0287
0.0281
0.0275
0.0268
0.0262
0.0256
0.0250
0.0245
0.0239
Between
mean and z
0.4713
0.4719
0.4725
0.4732
0.4738
0.4744
0.4750
0.4755
0.4761
If z-observed ≥ than 1.96
then reject H0
Our z-score for the sludge company is:
z  observed 
X 
X
X 
19 18
1



 2.50

4
0.4
N
100
So, we reject the Null hypothesis in favour of the alternative hypothesis
How big must a difference be, before we reject H0?
Two types errors can occur
Type 1 error: reject H0 when it is true. (Also known as
α, "alpha“ error). We think our experimental
manipulation has had an effect, when in fact it has
not.
Type 2 error: retain H0 when it is false. (Also known
as β, "beta“ error). We think our experimental
manipulation has not had an effect, when in fact it
has.
Any observed difference between two sample means
could in principle be either "real" or due to chance - we
can never tell for certain.
But Large differences between samples from the same
population are unlikely to arise by chance.
Small differences between samples are likely to have
arisen by chance.
Problem: reducing the chances of making a Type 1
error increases the chances of making a Type 2 error,
and vice versa.
Psychologists therefore compromise between the
chances of making a Type 1 error, and the chances of
making a Type 2 error:
We set the probability of making a Type 1 error at 0.05.
When we do an experiment, we accept a difference
between two samples as "real", if a difference of that
size would be likely to occur, by chance, 5% of the
time, i.e. five times in every hundred experiments
performed.
Directional and non-directional hypotheses:
Non-directional (two-tailed) hypothesis:
Merely predicts that sample mean ( X ) will be
significantly different from population mean (µ)
X < µ: Differences this
extreme (or more) occur
by chance with p = 0.025.
X > µ: Differences this
extreme (or more) occur by
 chance with p = 0.025.

X <µ
X =µ
possible differences
X>µ

Directional (one-tailed) hypothesis:
More precise - predicts the direction of difference
(i.e., either predicts X is bigger than µ, or predicts
X is smaller than µ).
X > µ: Differences this

extreme (or more) occur by
chance with p = 0.05.

X <µ
X =µ
possible differences
X>µ