The one sample t-test allows us to compare a sample mean to a population mean to determine if they are statistically
significantly different. It is the next step in learning about hypothesis testing.
The one sample t-test is a parametric procedure.
It tests whether a sample mean is statistically significantly different than a population mean
when the standard deviation of the population, or sigma, is unknown.
So, for example, a
population of polar bears is known to walk 100 miles per week.
Now that population would, of course, have a mean and a standard deviation,
but we do not know the standard deviation of the population
because we have not actually measured the entire population.
We estimated the mean from previous samples,
so we draw a sample of polar bears from this population and give that sample of polar bears caffeine.
We find that our caffeinated sample of polar bears walks an average of
150 miles per week with a standard deviation of 9 miles.
We could use that standard deviation of 9 to estimate the unknown
standard deviation of the population and then determine if the mean difference between 100 miles and
150 miles is statistically significantly different.
If it is, we would credit the caffeine for causing the difference.
Let's return to the research design example that we first learned about when we studied the z-test.
We are testing the research question,
"Is recall better under hypnosis or without?"
To study this we find a memory test that reports the mean scores for people who are not under hypnosis.
We administer this same memory test to a sample of people who are under hypnosis.
Therefore, the independent variable is the condition whether or not the subjects are under hypnosis.
The dependent variable is memory, the number of words recalled.
By comparing the average number of words recalled from our hypnotized sample to the population
value of unhypnotized test takers we could answer the question of whether
hypnotized participants remember more words than the
unhypnotized population.
When using SPSS to conduct a one sample t-test you need one categorical,
dichotomous, independent variable and one continuous or scale
dependent variable.
The categorical independent variable creates two groups,
and the dependent variable is measured and compared between the groups.
Now all inferential statistics are built on certain assumptions. If your assumptions are wrong then your conclusions will be wrong.
So it is important that you check your assumptions.
The first assumption that must be true in order to use a one sample t-test is that the independent variable must be
categorical and a single sample. For example, we have a single
randomly selected sample of polar bears.
All members of the sample belong to the category Polar Bears.
The sample is assumed to have been randomly selected.
We use a random sampling technique to choose the participants,
however, this type of test can also be done with convenience sampling.
The dependent variable must be scale level. The dependent variable must be
quantitative at the scale level or
likert scale data with at least five answer options.
The scale level data for the dependent variable is established when you choose your research methodology.
You should choose a data collection instrument survey or measurement that will give you scale level data.
For example, miles walked is ratio level data, so it is on a scale.
The one sample t-test assumes that you do not have extreme outliers in your dependent variable.
If you do have outliers, you can fix that by removing them, or better yet, winsorizing them to the next highest
reasonable value.
The scale variable scores should be independent, meaning that the data are not related or correlated.
For example, data are independent when you randomly select
individual participants. You would not randomly select families of polar bears and then treat them as individuals,
pretending as though their scores were unrelated.
Walking with others in your family will influence
how far you walk, and how far you walk might influence how far they walk.
Your walking would not be independent of others in the sample because
literally in this case your scores are related.
The sampling distribution of the dependent variable should be normally distributed.
So the dependent variable should be drawn from a population
whose sample means are normally distributed.
Assuming that the other assumptions are met this is not an issue. Especially when your sample size is greater than 30.
See my video on the central limit theorem for more details.
Here are the settings for null hypothesis significance testing.
The null hypothesis is that the sample mean equals
the population mean. So here is how you would write your null hypothesis
H0: μ = μ0
Where μ0 is the population mean.
You will literally plug in the population value where it says μ0.
The alternative hypothesis would be H1: μ = H0
Again, plugging in the population value where it says μ0.
The alpha level is typically set to α = .05
But it could be set to .01 or any other level chosen by the researcher.
The degrees of freedom is N - 1
Or the total sample size minus one.
The critical value is determined based on degrees of freedom and alpha. You will look up the critical value in
Student's t-Distribution Table, which is available on the last page of your notes.
So you can assume that the t-test is α = .05
two-tailed test unless you are told otherwise.
This is what you need to know, so let's practice with the one sample t-test.