A one sample test because the same group is being

a one-sample test because the same group is being tested twice (or sometimes
related/dependent groups e.g. pairs of twins)
25
• If we reject ܪ
଴
when it is true, this is a type I error
• If we fail to reject ܪ
଴
when it is false, this is a type II error
26
Types of error

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14
Central Limit Theorem
• If the population sampled from is Normally distributed, the ܺ
ത
distribution will be Normally distributed whatever the size of ݊
 Mean of ܺ
ത
distribution is
ߤ
 Standard deviation of ܺ
ത
distribution is
ఙ
௡
(also called ‘standard error’)
 Variance of ܺ
ത
distribution is
ఙ
మ
௡
 ࢄ
ഥ
~ࡺ(ࣆ,
࣌
૛
࢔
)
• Central Limit Theorem : if we take many random samples of size ݊ and
if ݊ ≥ 30 , the ܺ
ത
distribution will approximate a Normal distribution
even when the population sampled from is not Normally distributed
 s can be substituted in place of ࣌ so that
௦
௡
replaces
ఙ
௡
in calculations
27
Distribution of the sample mean
• When ݊ ≥ 30 and sampling is random, sample means (
ܺ
ത
) will be
Normally distributed (Central Limit Theorem)
 test statistic is
ݖ =
௫̅ିఓ
഑
೙
ൗ
( =
௦௔௠௣௟௘ ௠௘௔௡ି௛௬௣௢௧௛௘௦௜௦௘ௗ ௠௘௔௡
௦௧௔௡ௗ௔௥ௗ ௘௥௥௢௥
)
 can substitute s instead of ࣌ and replace ݖ =
௫̅ିఓ
഑
೙
ൗ
with ݖ ≈
௫̅ିఓ
ೞ
೙
ൗ
 test statistic is now Z, hence conducting a Z -test (using Normal dist)
• When ݊ < 30 , can also substitute s instead of ࣌ and use
appropriate t -distribution
 but only if sampling from a Normally distributed population
• As n increases (and therefore ݂݀ increases), the t -distribution
approaches the Normal distribution
 so t -tests and Z -tests give increasingly similar results
28

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15
Using a sample mean or median to test a hypothesis about
population central location,
when ࣌ is unknown and sampling is random
is sample
size large?
is sampling
from Normally
dist.
population?
no
no
yes
(CLT)
Z -test (can also use t -test)
Use s instead of ߪ
Sign Test
yes
29
t -test
Use s instead of ߪ
Section C
(Weeks 8 – 11)
Two variable analysis
30

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16
Chi-square test
(2 cat. variables)
• When working with two variables, our motivating question will
generally be “is there an association between the variables?”
• The chi-square (χ
ଶ
) test is used to detect the presence of an
association between two categorical variables
• The χ
ଶ
distribution is asymmetrical and skewed to the right
• Expected values are calculated by multiplying the appropriate row
total by the column total and dividing by the total number of
observations
ܧ =
ோ×஼
௡
• Hypotheses when performing a chi-square test will be:
ܪ
଴
: there is no association between variables
ܪ
ଵ
: there is an association
31
Chi-square test
• If there were perfect agreement between the observed and expected frequencies,
then we would expect the ߯
ଶ
test statistic to equal zero
• The more the observed and expected frequencies disagree, the greater the value
of the test statistic
• A ߯
ଶ
test is a one-tailed test
• The degrees of freedom for the ߯
ଶ
test are calculated as
݂݀ = ܴ − 1 × ܥ − 1 ,
where ܴ is no. of rows and ܥ is no. of columns
• Necessary conditions for you to check when conducting a ߯
ଶ
test are
1. observations must be independent
2.