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Hypothesis Test for Means
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For small samples (less than about thirty) and the process conforms to a normal distribution use the t-test. Choices are:
One Sample t-test
Two sample t-test
Paired t-test
For large samples, or where the process standard deviation is known and the process conforms to a normal distribution the choices are
One sample z-test
Paired z-test
Nonparametric tests for the mean include:
Wilcoxon Signed Rank Test (nonparametric equivalent of the one sample t-test)
Mann Whitney Test (nonparametric equivalent of the two sample t-test)
Use ANOVA for more than two samples.
A hypothesis test used to test the mean of a small sample taken from a population with a normal distribution against a specified value. The hypothesis:
H0 the population mean equals a specified value
H1 the popular mean is [equal to/less than/greater than] a specified value
The test is:
where:
 is the sample mean
m0 is the specified value
s the sample standard deviation
n the sample size
The critical value of the t statistic t0 can be found in t distribution tables, or the p-value can be found using the Excel function:
=TDIST(|t0|, n, Tails)
The number of degrees of freedom 'n' is n - 1, the number of tails depends on whether it is a one or two sided test.
The t-test requires that the population conforms to a normal distribution.
The paired t-test is used when the test units can be paired. For example, the blood pressure of five patients was tested before and after a medication:
| Patient |
Before
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After
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D
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| Andrew |
120
|
110
|
10
|
| Bill |
135
|
115
|
20
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| Charles |
110
|
110
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0
|
| David |
140
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135
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5
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| Eric |
115
|
110
|
5
|
A Two Sample t test would show no significant difference because the standard deviations of the 'before' and 'after' columns is high.
The paired test involves carrying out a One Sample t-tests on the differences.
See Hypothesis Test for Means
A hypothesis test used to to compare the means of two reasonably small (30 or less) samples to see if it is feasible that they come from the same population. The hypothesis is:
H0 the population means are equal
H1 the popular means are different
The test is:
Step 1: calculate the pooled standard deviation
Step 2: calculate the t statistics
The degrees of freedom:
n1, n2 sample sizes
s1, s2 sample standard deviation
A hypothesis test used to to compare the means of two samples to see if it is feasible that they come from the same population. The test is used with large samples (greater than 30) taken from a normal distribution. The hypothesis is:
H0 the population means are equal
H1 the popular means are different
The test is:
The p-value can be obtained from Excel using the function:
one-tail test: = 1 - NORMSDIST(Z0)
two-tail test: = (1 - NORMSDIST(Z0))/2
Alternatively the critical values of the z statistic can be found from tables. For a one sided test:
| a |
0.10
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0.05
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0.025
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0.01
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| Za |
1.28
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1.64
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1.96
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2.33
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A hypothesis test used to test the mean of a large sample (greater than about 30) against a specified value. The hypothesis:
H0 the population mean equals a specified value
H1 the popular mean is [equal to/less than/greater than] a specified value
The test is:
where:
is the sample mean
m0 is the specified value
s the sample standard deviation
n the sample size
The p-value can be obtained from Excel using the function:
one-tail test: = 1 - NORMSDIST(Z0)
two-tail test: = (1 - NORMSDIST(Z0))/2
Alternatively the critical values of the z statistic can be found from tables. For a one sided test:
| a |
0.10
|
0.05
|
0.025
|
0.01
|
| Za |
1.28
|
1.64
|
1.96
|
2.33
|
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