Psy 1191 Intro to Research Methods
Dr. Launius
1
Psy 1191 Statistical Methods Workshop: t-tests for 1 sample
Introduction: When analyzing data, you may have to decide if a sample mean is different from a
hypothesized population mean. To do this, you must have measurement data, calculate the mean and
standard deviation for your sample, and know something about the population.
For example:
You hear that the average person sleeps 8 hours a day. You think college students sleep less. You ask
20 college kids how long they sleep on an average day. You get the data and the mean sleep time is 6.5
hours. Is this luck? Did you happen to pick a group of light sleepers by chance? Or do college kids really
sleep less? What you are asking is whether college kids come from a population separate from the rest
of society. "Society" supposedly is all the people who contributed to the mean of 8 hours a night that we
hear about. You might recognize this example from Workshop #4 when we introduced the idea of
hypothesis tests. Here we are going to concentrate on the test statistic.
Does Your Group Come from a Different Population than the One Specified? The problem is that if
you calculate a sample mean and it is different from the one hypothesized, there are two possible
reasons for the difference: Your sample comes from a different population and the sample mean
represents a different population mean. When this happens, you reject the Null Hypothesis. The group
comes from the same population and the mean varies by chance. You just happened to pick a sample
group with a mean that misrepresents the population it came from. The group isn't really different. When
this happens you fail to reject the Null Hypothesis
The statistic you need to decide whether to reject or fail to reject the null hypothesis is the one sample ttest. This statistic compares the sample mean to the population mean and gets an estimate of the
probability that the sample mean is different by chance. First, you will calculate the mean and standard
deviation for your group. You might use a computer to do all of this for you. Let me introduce some
notation. The sample mean is usually noted as
The population mean is noted as
to see whether or not
-test formula to compute the t-value.
Let’s look at our chart again. You pull an infinite set of sample means & make a distribution of these
means (Sampling Distribution of the Mean). The mean of this distribution is μ if the Null Hypothesis is
true. Its width is measured by the standard error of the mean (dark gray arrow from mean to curve). You
calculate a t-value & see if it is in the tail. If the t score is far away: Reject the Null Hypothesis
From Wadsworth Publishing:
http://www.wadsworth.com/psychology_d/templates/student_resources/workshops/res_methd/
Psy 1191 Intro to Research Methods
Dr. Launius
2
Psy 1191 Statistical Methods Workshop: t-tests for 1 sample
The t-score is like a z-score and tells you if the sample mean is far away from the population mean (μ).
You will use the sampling distribution of the mean to do this. If the t-value is big enough (you look it up in
a t-table), you will reject the Null Hypothesis and say you have a difference. If the t-value is not big
enough you will say that you have not found a difference and fail to reject the Null Hypothesis. How does
the t ratio do this? The t-test sets up a sampling distribution of means with a mean that is specified by
the null hypothesis ( ). The t-test calculates if your sample mean is far in the tails of the sampling
distribution and far away from the population mean ( ). If it is, the difference may be too big to be
chance. If your group was really from the null hypothesis distribution, then the difference (and your tscore) should be close to zero.
The t-score is made a relative score by dividing the difference between the sample mean and m by the
standard error of the mean. See your book for the standard error formula. That's how we know if your
sample mean is rare by chance as we can calculate the areas in the tails.
Here's an example:
 The average sleep time is supposed to be 8 hours a day ( ).
 We think college students sleep a different amount, maybe more - maybe less.
 We survey 10 students to see how much they sleep.
 The data are as follows (each cell represents a student):
From Wadsworth Publishing:
http://www.wadsworth.com/psychology_d/templates/student_resources/workshops/res_methd/
Psy 1191 Intro to Research Methods
Psy 1191 Statistical Methods Workshop: t-tests for 1 sample
Here's the output from a computer package:
Why do you think this data set is not significant?
Answer the following questions about this test.
1). What is the value of the sample mean?
2). The print out refers to a “test value” of 8; what is this in our hypothesis test?
3). What does the “mean difference” of –2.8 refer to?
4). Do we reject or fail to reject the null hypothesis? Explain
Here's another example:
Here's the output from a computer package:
Why do you think the dataset is not significant?
From Wadsworth Publishing:
http://www.wadsworth.com/psychology_d/templates/student_resources/workshops/res_methd/
Dr. Launius
3
Psy 1191 Intro to Research Methods
Psy 1191 Statistical Methods Workshop: t-tests for 1 sample
Bottom Line:
 You get the data.
 You calculate the mean and standard deviation.
 You calculate the t ratio.
 You look it up.
 If the probability of t is less than .05 or .01, you have significance.
 You conclude the group is different from the Null Hypothesis mean.
REMEMBER THIS:
 Big t (absolute value)
 Small p
 Small p = significance
From Wadsworth Publishing:
http://www.wadsworth.com/psychology_d/templates/student_resources/workshops/res_methd/
Dr. Launius
4