INTRODUCTION :
A statistical examination of two population means. A two-sample t-test examines whether two samples are different and is commonly used when the variances of two normal distributions are unknown and when an experiment uses a small sample size. For example, a t-test could be used to compare the average score obtained by Class A in Maths team to the average score obtained by Class B in the same.
Z-TEST
A statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic is assumed to have a normal distribution and nuisance parameters such as standard deviation should be known in order for an accurate z-test to be performed.
A singe sample
experiment compares one sample to a population. There are two types of
statistics you can use to compare a single sample to a population:
If your sample size is above 1000 then this is the appropriate statistic to
use.
The single sample z-test formula is shown below:
The formula reads: the z-test equals the sum of
the sample mean minus the population mean and then divided by the standard
error.
As mentioned in an earlier lesson, if we do not
know the population standard deviation we can use the sample standard deviation
to estimate the standard error.
The formula reads: the standard error equals the
sample standard deviation divided by the square root of the sample size.
When we use an alpha level of 0.05, any z score
that results in a probability of less than 0.05 allows us to reject the null
hypothesis and accept the research hypothesis. All you need to know is
the minimal z score necessary for significance. Rather than
constantly going to the Z table you can just memorize the one-tailed and
two-tailed z scores that equate to a 0.05 level of significance. If you go back
to the end of Lesson 11 you will see that a two-tailed hypothesis needs
a z score of 1.96 to be significant while a one-tailed
test needs a z score of 1.64 to be significant.
If your sample size is below 1000 then this is the appropriate statistic to
use.
The single sample t-test formula is shown below:
The formula reads: the t test equals the sum of
the sample mean minus the population mean and then divided by the standard
error.
As with the z-test above, if we do not know the
population standard deviation we can use the sample standard deviation to
estimate the standard error.
The formula reads: the standard error equals the
sample standard deviation divided by the square root of the sample size.
The t distribution is
similar to the z distribution in that both are symmetrical, bell-shaped
sampling distributions. The overall shape of the t distribution is influenced
by the sample size used to generate it. Therefore, when the sample is
large (n >1000) you should use the z-test and when the sample is small you
should use the t-test. Because of this fact we need to use degrees of
freedom to determine our significance threshold. For a single sample t-test the
degrees of freedom calculation is as follows:
df = n - 1
Now we can go to the T
Table to see if our statistic is significant.
|
|
One-Tail
= .4
|
.25
|
.1
|
.05
|
.025
|
.01
|
.005
|
.0025
|
.001
|
.0005
|
|
df
|
Two-Tail
= .8
|
.5
|
.2
|
.1
|
.05
|
.02
|
.01
|
.005
|
.002
|
.001
|
|
1
|
0.325
|
1.000
|
3.078
|
6.314
|
12.706
|
31.821
|
63.657
|
127.32
|
318.31
|
636.62
|
|
2
|
0.289
|
0.816
|
1.886
|
2.920
|
4.303
|
6.965
|
9.925
|
14.089
|
22.327
|
31.598
|
|
3
|
0.277
|
0.765
|
1.638
|
2.353
|
3.182
|
4.541
|
5.841
|
7.453
|
10.214
|
12.924
|
|
4
|
0.271
|
0.741
|
1.533
|
2.132
|
2.776
|
3.747
|
4.604
|
5.598
|
7.173
|
8.610
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5
|
0.267
|
0.727
|
1.476
|
2.015
|
2.571
|
3.365
|
4.032
|
4.773
|
5.893
|
6.869
|
|
6
|
0.265
|
0.718
|
1.440
|
1.943
|
2.447
|
3.143
|
3.707
|
4.317
|
5.208
|
5.959
|
|
7
|
0.263
|
0.711
|
1.415
|
1.895
|
2.365
|
2.998
|
3.499
|
4.029
|
4.785
|
5.408
|
|
8
|
0.262
|
0.706
|
1.397
|
1.860
|
2.306
|
2.896
|
3.355
|
3.833
|
4.501
|
5.041
|
|
9
|
0.261
|
0.703
|
1.383
|
1.833
|
2.262
|
2.821
|
3.250
|
3.690
|
4.297
|
4.781
|
The T Table continues
The T Table is similar
to the R Table we used in lesson 7. The degrees of freedom are in the far left
column and the levels of significance for each type of tailed test are in the
above column headings. As with the R Table critical R values, the T Table gives
you the critical T values. Your calculated T value must surpass the critical T
value for your statistic to be considered significant.
III. Two Sample
Experimental Statistics
For these experiments we
are comparing two samples. This is the very common control group vs.
experimental group research design. There are two ways to conduct the analysis
base on your sample groups.
If your two sample groups are independent of
each other then you can conduct a t-test for independent groups. The formula
for this specific type of t-test is as follows:
The formula reads: t (for independent groups)
equals the sum of sample mean number 1 minus sample mean number 2 and then
divided by the standard error of the difference.
The standard error of the difference is
similar to the standard error calculated earlier. It simply is a better
estimate for two independent samples. The standard error of the difference
between independent sample means can be calculated with the formula below:
The formula reads: the standard error of the
difference equals the square root of the standard error of sample one squared
plus the standard error of sample 2 squared.
The calculation for the degrees of freedom is as
follows:
df independent groups = (n1 -
1) + (n2 - 1)
Once your calculations are complete you go to
the T Table to see if your statistic is significant as above.
If the two samples are
not independent of each other but instead are positively correlated to each
other, we conduct a t-test for correlated groups. There are two ways of
calculating this statistic. One uses the correlational coefficient (r) of the
two samples and one does not.
1. t-test for Correlated Groups: using the
r value
The t-test formula is the same as was used for independent groups:
The formula reads: t (for correlated groups)
equals the sum of sample mean number 1 minus sample mean number 2 and then
divided by the standard error of the difference.
The
new standard error formula is as follows:
The formula reads: the standard error of the
difference equals the square root of the following: the sum of the squared
standard error of the first sample mean and the squared standard error of the
second sample mean. Then subtract the product of 2 times the r value times the
standard error from the first sample times the standard error from the second
sample.
The calculation for the degrees of freedom is as
follows:
df correlated groups = number of pairs - 1
Once your calculations are complete you go to
the T Table to see if your statistic is significant as above.
2: t-test for correlated samples: using raw data
The t-test for correlated groups using the raw data is as follows:
The formula reads: t
(for correlated groups) equals D bar divided by the standard error of the
difference.
D bar is the mean of all
the difference scores. Difference scores are calculated by subtracting each Y
value from its X pair value. You then sum these difference scores and divide by
the number of pairs to get D bar. An example is shown in the table below:
|
X
|
Y
|
D
|
|
15
|
5
|
10
|
|
7
|
1
|
6
|
|
12
|
8
|
4
|
|
18
|
12
|
6
|
|
8
|
9
|
-1
|
|
|
|
sum
= 25
|
|
n
= 5
|
D
bar = 25/5
|
D bar = 5
|
The
new standard error formula is as follows:
The formula reads: the
standard error of the difference equals the square root of the following: D bar
squared subtracted from the sum of D squared over n and then this entire sum
divided by the number of pairs minus one.
In order to get the sum
of D squared you need to generate a new column of data as is shown below:
|
X
|
Y
|
D
|
D2
|
|
15
|
5
|
10
|
100
|
|
7
|
1
|
6
|
36
|
|
12
|
8
|
4
|
16
|
|
18
|
12
|
6
|
36
|
|
8
|
9
|
-1
|
1
|
|
|
|
sum
= 25
|
sum = 189
|
|
n
= 5
|
D
bar = 25/5
|
D
bar = 5
|
|
The calculation for the degrees of freedom is
the same:
df correlated groups = number of pairs - 1
Once your calculations
are complete you go to the T Table to see if your statistic is significant as
above.
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