UNDER CONSTRUCTION |
Your textbook:
Chapter 10: Hypothesis tests with two independent samples
Let's start with a brief review.
In chapter 8 we looked at ways to test whether a sample was probably from a known population (that is we know the pop s). In chapter 9 we looked at the same situation as in 8, but the situation is one in which we don't know the population s, so we need to use an estimate.
The logic of this chapter should seem similar to the last two, but things get a little more complicated, because of the situation that we are interested in. Now we are going to look at a situation where we are interested not in a single population, but at two different populations. And again, we'll deal with situations in which we don't know the s for either of these populations, so we'll have to use estimates.
An experiment that uses a separate sample for each treatment condition (or each population) is called an independent-measures research design. Often
you'll also see it referred to as a between-subjects or between-groups desgin.
So we'll use the same logic and steps for hypothesis testing that we used in the previous chapter, and fill in the details of the differences as we go.
step 1: state your H0 and H1 & figure out your critera: a = ?
step 2: figure out if your test is one-tailed or two-tailed
step 3: figure out the df for your test
step 4: find the critical t-score from the table
step 5: compute your t-score for your sample
step 6: compare your t-score with the critical t-score
step 7: make your conclusions about the H0
Let's start with step 1.
Figuring out your critera is exactly the same process as before, you pick what your field has decided as being an accepted level of alpha (chance of making a type I error). For our example, let's assume a = 0.05
The hypotheses are going to be a bit different looking, because the situation is different. Remember, that now we are making hypotheses about two different populations.
For example, suppose that you want to compare two different treatments (e.g., two ways of studing, two different drugs, etc), or you want to compare two groups of people (e.g., men vs. women, young vs. old, etc.).
So now, the hypotheses are about population A (men) and population B (women), and how they are different from one another. Suppose that we are interested in how tall men and women are.
So the H0 hypothesis would be that men and women are the same height.
That is:
Our alternative hypthesis could be that men and women are different heights.
That is:
Step 2.
Okay, is this a one-tailed or two-tailed hypothesis?
What might the hypothesis be for a one-tailed test?
Men are taller than women. H0: mA = mB & H1: mA > mB
Step 3.
What are the degrees of freedom? Well we need some more information about our example before we can answer this questions. Let's start with this conceptually, then fill in the necissary details in our example.
We are going to be using two samples, one to represent each population.
Remember, that because we're using samples, we can only estimate the values of the population parameters and so we're going to need to take degrees of freedom into account.
Any guesses as to how we'll compute our df?
Think about it this way, with one sample we used
n - 1 because all of the values in the sample are free to vary but one, because we know the value of the sample mean.
Now consider the current situation. We've got two samples. How many values are free to vary?
sample 1: n A- 1
sample 2: n B- 1
so together there are nA + nB - 2 = df
Okay, so additional information do we need for our example?
we need to know how many individuals we have in our samples.
This is a good time to look at the actual data (from the data collected at the begining of the semester)
men's heights: 67, 73, 74, 70, 70, 75, 73, 68, 69
women's heights: 69, 63, 67, 64, 61, 66, 60, 63, 63
so what is nA? = 9
so what is nB? = 9
So the df for our example is: nA + nB - 2 = 9 + 9 - 2 = 16
Step 4.
So what is our critical t? Go to the table, look up the value for:
two-tailed, a = 0.05, df = 16.
tcrit = 2.12
Step 5.
Now comes what will look to be the big difference. We need to compute our observed t statistic. Basically, at the conceptual level, the formula is the same. However, at the practical level, it is a bit more complex because we have two samples, which means that we have two estimates. Let's break this formula into several parts.
conceptually: tobs =
in other words, we're intersted in the difference between the two populations, so to compute the t statistic we need to see if the difference between our two samples is different from the difference between the two populations.
So the numerator is pretty much straight forward:
= the difference between the two sample means
(mA - mB) = 0: remember that's the H0
The denominator is where things will look a bit more complex:
what is?
this is the an estimate of the error from the two samples. Recall that each sample will have some sampling error associated with it. What we need to do here is pool the error from the two samples.
- because each sample may be of different sizes (n's) we need to weight each sample's estimate of variability by its degrees of freedom.
pooled variance =
we can simplify the equation, recall that s2 = SS/df
so, by substituting SS for df(s2) we get:
pooled variance =
Okay, notice that we're not at yet. We still need to compute that.
Remember that the formula for estimated standard error of :
=
The formula for is similar:
=
So let's fill in the numbers from our example.
First we need to go back to the raw numbers and compute the SS's and the sample means.
Here are the results or those computations:
= 71.0 = 64.0
SSA = 64.0SSB = 66.0
sA = 2.83 sB = 2.87
So, = = 8.125
So, = == 1.34
Now let's put together the whole t statistic (finishing step 4)
tobs = = = 5.22
step 6.
So now we compare the two t statistics.
tobs = 5.22tcrit = 2.12
step 7:
Our observed (computed) t statistic is greater than the critical t statistic, so we feel confident in rejecting the H0. There does seem to be a difference between the heights of men and women.
What are the assumptions of our independent measures t test?
1) The observations are independent (as we discussed last chapter)
2) The two populations are normally distributed (also discussed last time)
** new **
3) The two populations have equal variances. This is referred to as homogeneity
of variance.
recall that in the formula we pool our sample variances. This is an okay thing to do if the variances are about the same.
However, it isn't okay if they are very different. How big a difference is too big? You can perform Hartley's F-max test for homogeneity of variance (see box 10.3, pg. 304).
There is also a less exact rule of thumb: for small samples
(n < 10), if one sample variance (s2) is more than four times bigger than the other, then homogeneity of variance is probably violated. For larger samples, if variance is 2 times larger, there is probably a violation.
If, time permits, another example (#10, page 309)
The effects of fatigue on mental alertness. Group 1 stays awake for 24 hours,
group 2 gets to go to sleep. Then tested to see how well they detect a light on screen.
two groups: n1 = 5, n2 = 10, 1 = 35, 2 = 24, SS1 = 120, SS2 = 270
H0: m1 - m2 ² 0 H1: m1 - m2 > 0
df = n1 + n2 - 2 = 5 + 10 - 2 = 13 and a = 0.05
tcrit = 1.771
= 30.0
tobs = = = 3.67
So we reject H0 since tobs falls in the region of rejection