, 
).We give the subjects intensive "Get Smart" training and then administer an IQ test. The sample mean IQ is 113 and the sample standard deviation is 10. Did the training result in a significant increase in IQ score?
Lesson 10 will consist of the following topics
For lesson 10, read pages 123-125, 131-134 in Practical Statistics for Educators,
Third Edition by Ruth Ravid (2005, University Press of America)
or read pages 294-303 in Basic Statistics for Behavioral Science Research
2nd ed by Mary B. Harris (1998, Allyn and Bacon)
or
read pages 187-190, 198-202 in Practical Statistics for
Educators, 2nd Edition by Ruth Ravid (2000, University Press of America)
or read pages 186-189 in Practical Statistics for Educators
by Ruth Ravid (1994, University Press of America).
In our last lesson we looked at the process for making inferences about research. In this context we looked at the significance of a single score. We wanted to see if a score differed significantly from a population value. To test statistical hypotheses involving a single score we calculated the scores Z-score. We referred to this as the Z-score test. As a reminder the formula for the Z-score (or the Z-score test) was
In this lesson we are going to move on and look at inferential statistics to test hypotheses concerned with comparing a single sample (instead of a single score) with some population parameter. We will discuss two statistics to use with single samples.
Research Problem: We randomly select a group of 9 subjects from
a population with a mean IQ of 100 and standard deviation of 15
(, ).
We give the subjects intensive "Get Smart" training and then administer
an IQ test. The sample mean IQ is 113 and the sample standard deviation is 10.
Did the training result in a significant increase in IQ score?
In this problem we see that we have a single sample and we wish to compare the sample mean with a population mean. We know what the population standard deviation is. From what we have said we can see that the inferential statistic we need to use here is the Z-test.
The formula for the Z-test is
Where 
is the sample mean (113 in our problem)
is the sampling distribution of the mean.
The sampling distribution of the mean is the mean of
a set of many many sample means taken from a population. It is the mean of
all the means. In practice the sampling distribution of the mean is the
same as the population mean, so we can use 
instead of
. In our problem the population mean is 100.
is the standard error of the mean.
It is the standard deviation of many many sample means. Unfortunately for us
the standard error of the mean does not equal the population
standard deviation but instead is equal to the population standard deviation (sigma)
divided by the square root of the sample size (n). So for our problem
We are now ready to calculate the value of Z for our problem. We have all the information we need:
So the value of Z for our problem is
We can now go ahead and complete the six step process for testing statistical hypotheses for our research problem.
Consider the following research problem: We have a random sample of 25 fifth grade pupils who can do 15 pushups on the average, with a standard deviation of 9, after completing a special physical education program. Does this value of 15 differ significantly from the population value of 12?
In this problem we are comparing a sample mean with a population mean but we do not know the population standard deviation. We can't use the Z-test in this case but we can use the one-sample t-test. The one sample t-test does not require the population standard deviation. The formula for the one-sample t-test is
Where 
is the sample mean,
is the population mean,
and 
is the sample estimate of the standard
error of the mean.
In the problem we are considering, we do not know the population standard deviation (or the standard error of the mean) so we estimate it from the sample data. The sample estimate of the standard error of the mean is based on S (the sample standard deviation) and the square root of n (the sample size).
If you look back at the research problem you will see that we have all the data we need to calculate the value of t.
The sample mean, is 15.
The population mean, is 12.
The sample standard deviation, S is 9.
The sample size, n is 25.
We can thus calculate the value of t as follows:
The t statistic is not distributed normally like the z statistic is but is distributed as (guess what) the t-distribution, also referred to as student's distribution. We will use this distribution when we do the six step process for testing statistical hypothesis. To use the table for the t-distribution we need to know one other piece of information and that is the degrees of freedom for the one sample t-test.
Degrees of freedom is a mathematical concept that involves the amount of freedom you have to substitute various values in an equation. For example say we have three numbers that add up to 44. For the first two numbers we are free to use any numbers we wish but when we get to the third number we do not have any freedom of choice if the sum is to be 44. Therefore we say that with the three numbers we have two degrees of freedom.
For the one-sample t statistic the degrees of freedom (df) are equal to the sample size minus 1, or for our research problem:
To put all this information together go ahead and look at the example problem using the one-sample t-test.
Please send electronic mail to the course instructor if you have any questions about this lesson or other concerns.