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Tests of Significance
L1
Probability: is a numerical measure of the likelihood that an event will occur
ⱴ The Z test,
ⱴ The t test, and
ⱴ The X² test
What is a test of significance?
A/ It is a formal procedure for comparing observed data with a hypothesis whose truth we
want to assess.
The results of tests are expressed in terms of a probability that measure how well the data
and hypothesis agree
Stating Hypothesis
ⱴ A hypothesis is a statement about parameters in the population, ex: µ1= µ2
ⱴ Hypotheses are only concerned with the population
Null hypothesis (Ho)
ⱴ A statistical test begins by supposing that the effect, we want, is not present.
This assumption is called the null hypothesis
ⱴ Then we try to find evidence against this claim (hypothesis)
ⱴ Typically, Ho is a statement of “no difference” or “no effect”
ⱴ We also want to assess the strength against the null hypothesis
Alternative Hypothesis (Ha)
ⱴ It is the statement about the population parameter that we hope or suspect is
true (i.e. what we are trying to prove or the effect we are hoping to see)
ⱴ Ha is a statement of difference or relationship
ⱴ It can be one tailed (< or >) (ex: Ha > Ho) or two tailed (< and >) (ex: Ha µ1≠ µ2)
Types of statistical tests:
ⱴ Parametric tests: assume that variables of interest are measured on interval
scale or ratio scale, usually continuous quantitative variable. There is assumption
that variables are normally distributed
ⱴ Non parametric tests: assumed that the variables are measured on a nominal or
ordinal scale
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Steps of hypothesis testing:
ⱴ State the null hypothesis
ⱴ State the alternative hypothesis
ⱴ State the level of significance
ⱴ Choose the correct test statistics
ⱴ Computed the test statistics
ⱴ Determine the critical value of a statistics (needed to reject the Ho)
from a table of sampling distribution values
ⱴ Compare computed to critical value
ⱴ Accept or reject the Ho.
Significance level:
ⱴ Usually, it is represented as α
ⱴ It is the value of probability below which we start consider significant differences
ⱴ Typical levels used are 0.1, 0.05, 0.01 and 0.001
ⱴ The usual alpha level considered in medicine is 0.05
The Z test
One sample Z – test
ⱴ That of one sample mean:
o Steps for testing one sample mean (with σ known), irrespective of sample
size
ⱴ State the Ho (Ho: µ1= µ2)
ⱴ State the H1 (H1: M1≠ M2)
ⱴ State the level of significance (example 0.05)
ⱴ Calculate the test statistics:
Z =
n
Mo
X
/
5. Find the critical value
a. for Z= 1.96
= 0.05
a. for Z= 2.58
= 0.01
6. Decision:
Reject Ho if test statistics > critical value i.e. P value < the significance level
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7. State your conclusion:
ⱴ If Ho is rejected, there is significant statistical evidence that the population mean
is different than the sample mean
ⱴ If Ho is not rejected, there is no significant statistical evidence that the
population mean is different from the sample mean
ⱴ Z – test for sample proportion:
Z =
n
P
)
1
(
Z- Test for differences between 2 means:
Z=
)
(
2
1
2
1
2
1
)
(
)
(
X
M
M
X
X
2
1
X
X
=
2
2
1
2
n
n
Testing the difference between 2 sample proportions:
Z =
2
1
2
1
2
1
)
(
)
(
P
S
P
P
P
Where Sp1-p2 =
]
1
1
)[
1
(
2
1
n
n
p
P
P (Pooled)=
2
1
2
2
1
1
n
n
P
n
P
n
T-test
One Sample T-test
ⱴ In small sample size, when σ is not known, the sample standard deviation is used to
estimate σ and the Z-statistics is replaced by the T-statistics.
ⱴ t= x - µ
o S/ √n
ⱴ When the x is the mean of a random sample of size n from a normal distribution
with mean µ, then t has a student t-distribution with n-1 degree of freedom (df)
ⱴ The df is the number of scores in a sample that are free to vary
ⱴ The df is a function of the sample size determines how spread of the distribution is
(compared to the normal distribution)
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The T-distribution
ⱴ Example, using the normal curve, 1,96 is the cut-off for a two tailed test at the
0,05 level of significance
ⱴ On a t-distribution with 3 df (a sample size of 4), the cut-off is 3.18 for a 2-tailed
test at the 0.05 level of significance
ⱴ If your estimate is based on a larger sample of 7, the cut-off is 2.45, a critical
score closer to that for the normal curve
ⱴ The t-distribution is a bell-shaped and symmetrical one that is used for testing
small sample size (n < 30)
ⱴ The distribution of the values of t is not normal, but its use and the shape are
some what analogous to those of the standard normal distribution of z.
ⱴ T spreads out more and more as the sample size gets small.
ⱴ The critical value of t is determined by its df equal to n-1
Finding tcrit using t-table
ⱴ T-table is very similar to the standard normal table
ⱴ The bigger the sample size (or df), the closer the t-distribution is to a normal
distribution
T-test for two sample means
ⱴ t= lx1-x2l
o SE(x1-x2)
ⱴ SE(x1-x2) = Spooled * √ 1 + 1
n1 n2
2 2
ⱴ Spooled = S1 (n1-1) + S2 (n2-1)
n1 + n2 – 2
N.B df for 2 sample means in t-test = n1+ n2- 2
The X² Test
χ² = ∑
E
E
2
)
0
(
E=
Trc
Tc
Tr
df= (r- 1) (c - 1)
Mubark A. Wilkins