disc 11
martha04
M.Chapter11Salkind_ACC11.pptxChapter Eleven (Salkind)
t(ea) For Two:
Tests Between The Means Of Different Groups
One IV With Two Levels
One caveat before we move deeper into this chapter …
When we discuss studies that have two groups, students often get a little confused and think we are looking at two different independent variables. This is a mistake!
When looking at two group designs in this chapter, we are still focusing on one independent variable—it just happens to have two levels. For example, “Condition” might be our single IV, but the two levels in “Condition” include 1). the experimental group and 2). the control group. To put it another way …
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Any IV Has At Least Two Levels
An independent variable, almost by definition, involves the comparison of at least two different levels. Take the control group and the experimental group design …
If the “experimental” group is its own IV, then what are you comparing it to?
You may be tempted to say “the control group”. But if you mistakenly think the control group is a second IV, then what are you comparing that group to?
Circular argument, right! No, we have one IV with two levels (experimental versus control)
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More Complex Research Designs
Of course, this doesn’t mean that we can’t have a second IV in a research study. In fact, lots of studies look at more than one IV at a time, something you will see more next semester when you begin Research Methods and Design II. In that class, you will actually run a 2 X 2 factorial study (two IV’s with two levels each, or four conditions)!
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What About More Than Two Levels?
We could also have a single independent variable that has three levels (maybe two different control conditions and one experimental condition), four levels, or a hundred levels!
Just keep in mind that the number of IVs and number of levels for the IV are two different concepts
For our present Chapter 11, though, we are only going to focus on one IV, and our one IV has two levels (or two groups total)
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Good & Bad News
Here’s the bad news …
We have another formula, a looooong one, but one we will figure out together
Here’s the good news!
We see the SPSS t-Test version, too! Yes, that’s good news
We will once again use our 8 steps for significance (from Chapter 9) to see how two groups differ from one another
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Chapter Outline
In this chapter we cover the following items …
Part One: Introduction To The t-Test For Independent Samples
Part Two: Computing The Test Statistic
Part Three: Special Effects: Are Those Differences For Real?
Part Four: Using The Computer To Perform A t-Test
Part Five: An Eye Toward The Future
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Part One
Introduction To The t-Test For Independent Samples
Trick or Treat!
Imagine you are twelve years old, trick-or-treating for what may be the last time before you are “too mature for that nonsense”
You’re a little greedy with the candy. You’re twelve, so the more candy the better, right?
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What Would You DO?
You approach a house where no one is obviously at home, but the front porch light is on and you happen to see a pot of gold! Literally – a pot of golden (chocolate) coins.
No one is around. It’s you and that candy. There’s a sign that says “Please take no more than three pieces.” Do you comply, or do you dig in like the greedy pirate you might be dressed as?
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Let’s Test This Question!
Let us set up a couple of different study groups, much like a researcher name Gibbons once did
Group #1: There is a large mirror right behind the candy bowl (plus the note!). This way, the trick-or-treater will see themselves in the mirror (which might make them pause in deciding to break the “take only three pieces of candy” rule)
Group #2: There is no mirror. It’s just a bowl of candy with the note pleading to take no more than three pieces
Who do you think will break the “three piece” rule more often?
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What Kind of Study Design?
Let’s take a step back and figure out what kind of study design we need in order to look at our two trick-or-treater conditions
1. Are you examining relationships between variables or are you examining the difference between groups?
In Chapter 11, we are looking at two different groups, not variables, so we choose our between groups option
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Are Participants The Same?
2. Next, we must ask ourselves, “Are the same participants being tested more than once?”
If yes, it will lead us to a test that deals with dependent samples (which we will get to in Chapter 12, Salkind)
If no (which is our answer here), we have one more question to ask: How many groups are we dealing with?
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How Many Groups?
3. How many groups are we dealing with?
If we are looking at only two groups (which we are), we will use a t-Test for independent samples
If we are looking at more than two groups (at least three minimum), we will use an analysis of variance (ANOVA)
Don’t worry – we’ll get to this one in Chapter 13!
These decisions might be easier with a handy flow chart …
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This is where we are in Chapter 11
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Pop-Quiz 1: Quiz Yourself
The t-Test for independent means is used when there are ____________________.
A). only two groups in total
B). only one group
C). two groups or more
D). one group or more
Answer 1: A
The t-Test for independent means is used when there are ____________________.
A). only two groups in total
B). only one group
C). two groups or more
D). one group or more
Pop-Quiz 2: Quiz Yourself
If you want to examine the difference between the average scores of three unrelated groups, which of the following statistical techniques should you select?
A). Regression
B). Dependent samples t test
C). Analysis of variance
D). Independent samples t test
Answer 2: C
If you want to examine the difference between the average scores of three unrelated groups, which of the following statistical techniques should you select?
A). Regression
B). Dependent samples t test
C). Analysis of variance – We’ll get to this one in our later!
D). Independent samples t test
Assumption # 1
All statistical tests have assumptions, and the t-Test is no different. Here are the three assumptions to keep in mind …
First, t-Tests assume a homogeneity of variance between the two groups in the study
This is just a fancy way of saying that the amount of variability in one group is similar to the amount of variability in the other group
The larger the sample size, the more likely variance will be homogenous (this is something we will discuss more next semester in Research Methods and Design II)
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Assumption # 2
All statistical tests have assumptions, and the t-Test is no different. Here are the three assumptions to keep in mind …
Second, t-Tests assume that we are using data based on the mean (rather than the median or the mode). As long as we have scaled, continuous data (plus a normal curve for each condition), we can determine a mean (average), and thus we can use the t-Test
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Assumption # 3
All statistical tests have assumptions, and the t-Test is no different. Here are the three assumptions to keep in mind …
Third, in Research Methods at FIU, we will only look at t-Tests that have two groups, but there are other t-Tests—like a single samples t-Test—that compare one sample mean to something like the population mean (much like we did with the Z-test).
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Part Two
Computing The Test Statistic
Computing The Test Statistic: t -Test
Computing The Test Statistic – The Independent Samples t-Test
Ready for that loooooong formula? Here it is.
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This Formula Is Not As Hard As It Looks!
Let’s unpack this a bit now …
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Computing The Test Statistic: t -Test
Computing The Test Statistic – The Independent Samples t-Test
t-Test Formula
is the mean for group 1
is the mean for group 2
n1 is the number of participants in Group 1
n2 is the number of participants in Group 2
is the variance for Group 1
is the variance for Group 2
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Really, Don’t Worry!
Like I said, don’t panic. We’ve already done a lot of these calculations before (like variance), this just puts them into a slightly longer formula
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Study Design Summary
Before we start plugging in numbers, recall our study design
We have two conditions (one independent variable, though it has two levels)
In condition #1, we have a mirror behind the candy bowl
In condition #2, we do not have a mirror
In BOTH conditions, we have a sign that says “Take no more than three pieces of candy”
Our dependent variable is how many pieces of candy the trick-or-treaters actually take
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Step 1 (of 8)
Step One: State The Null And Alternative Hypotheses
These are easy, right:
Null hypothesis: The groups do not differ, Ho: µ1 = µ2
Alternative hypothesis: The groups differ, H1:
This hypothesis looks non-directional, right? we’re not predicting that they differ in a specific way, just that they’re different! Keep this in mind for later.
Why am I predicting the groups will differ? An important part of hypothesis building (which we will discuss a lot when you get to Research Methods and Design II) is relying on theory to help justify your hypotheses
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Theory to Support The Hypothesis
Social psychologist Gibbons used the theory of self-focusing situations to predict that people who are more aware of themselves “self-focus” more. When thinking about yourself, it’s easier to ask, “Would I be a good person if I did this?”
Research shows that de-individuated people (those who are not self-aware, like rioters caught up in the excitement of a mob!) are more likely to act against their own morals.
Halloween costumes can easily de-individuate people. That is, you lose your sense of “self” when in costume. But if you see yourself in a mirror, it might remind you of … you!
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Of course, this threat might be a good way to get obedience from trick-or-treaters, too!
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Steps 3 & 4 (of 8)
Step Two: Set The Level Of Risk
Again, pretty easy. In psychology we’ll stick with p < .05
Step Three: Select The Appropriate Test Statistic
We know from our nifty flowchart (and the fact that we are comparing only two groups on a dependent variable that relies on the mean) that we are using an independent samples t-Test
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Step 4 (of 8)
Step Four: Compute The Test Statistic Value
Okay, ready for the hard part?
First, we need our data set, so let’s see how many pieces of candy trick-or-treaters take (Warning: I am making this data up!). Let’s say we have 20 trick-or-treaters in each condition, so total n = 40 …
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Data for Groups 1 and 2
DV = Pieces of Candy Taken
| Group #1 – Mirror Present | Group #2 – Mirror Absent | ||||
| 5 | 7 | 5 | 7 | 6 | 7 |
| 4 | 3 | 3 | 8 | 8 | 6 |
| 5 | 2 | 3 | 12 | 9 | 3 |
| 4 | 2 | 4 | 11 | 9 | 8 |
| 3 | 3 | 3 | 8 | 10 | 4 |
| 3 | 8 | 4 | 7 | 13 | 5 |
| 4 | 5 | - | 5 | 14 | - |
| Mean = 80/20 = 4 pieces each | Mean = 160/20 = 8 pieces each |
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Computing The Mean
We must first compute the mean and the variance
The mean is easy.
For condition #1 (mirror present), we have 20 scores (20 participants) and a total of 80 pieces of candy taken. 80/20 = a mean of 4 pieces taken
For condition #2 (mirror absent), we have 20 scores and a total of 160 pieces of candy taken. 160/20 = a mean of 8 pieces taken
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Computing The Variance
We must first compute the mean and the variance
The variance you’re also familiar with calculating!
Do you recall our variance formula? If not, here it is!
Since you’re already very familiar with how to calculate the variance (and SD), I’m not going to go through the steps here. However, if you’d like a little reminder of how to do it, scroll allllll the way to the end of this lecture, and you can see the calculations worked out step by step.
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Computing The Test Statistic: t -Test
Computing The Test Statistic – The Independent Samples t-Test
Recall our t-Test Formula
is the mean for group 1 = 4
is the mean for group 2 = 8
n1 is the number of participants in Group 1 = 20
n2 is the number of participants in Group 2 = 20
is the variance for Group 1 = 2.32 (s1 or SD would be 1.52)
is the variance for Group 2 = 8.53 (s2 or SD would be 2.92)
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Plugging In Values
Note: This is the variance! If you have the SD, make sure to square it to get the variance for 2.32 and 8.53. Also, be sure not to accidentally square the variance again! The variance is already squared.
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Working Through The Formula
=
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What Now?
Phew, all done! Or are we? What does a t value of – 5.43 really tell us? Well, first let’s discuss that minus sign.
We can actually ignore the negative or positive sign in dealing with our t-Test statistic. Why?
The negative or positive sign for our t-value all depends on what we arbitrarily designate as “group 1” or “group 2”
In other words, just ignore the negative sign!
Now, on to step 5!
t = – 5.43
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Pop-Quiz 3: Quiz Yourself
In the formula that computes a t value, what does s12 represent?
A). Scores for group one
B). Mean for group one
C). Number of participants for group one
D). Variance for group one
Answer 3: D
In the formula that computes a t value, what does s12 represent?
A). Scores for group one
B). Mean for group one
C). Number of participants for group one
D). Variance for group one
Step 5 (of 8)
Step Five: Determine The Value Needed To Reject The Null
Our “obtained” t-Test statistic value is of course t = 5.43
We need to determine our “critical” value next
That is, what value does 5.43 need to be larger than in order to say that 5.43 is so rare that it would occur by chance in only 5% of the time?
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Using The T-Test Table
We actually have a t-Test table (much like our z Score table) where we can look up our critical value.
You can find this table in Appendix B.2. (Salkind),
However, you can find the exact same table in Appendix A of your Smith and Davis textbook
Yep, it is a table of standard t-values, so the table is identical no matter where you find it!
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Three Things We Need
You’ll need three pieces of information to find the critical value in the t-test table:
1. Your desired p values
2. Whether you’re doing a One-Tailed test or a Two-Tailed
3. You’ll also see df (degrees of freedom)
Let’s look at each of these …
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Choosing The P Value
1. There are tabled values for different p values
Salkind (and Smith & Davis) have t values for the p < .10 level, p < .05 level, and p < .01 level
We picked .05 in step 2, so we will use that!
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One or Two Tailed Test?
2. One-tailed tests versus two-tailed tests
The difference between the one-tailed versus two-tailed tests depends on whether you predict a specific outcome or a more general outcome
For a specific outcome (Group A will be higher than Group B, or Group A will be lower than Group B), the one-tailed is best. This is a directional test
Consider our Halloween candy study …
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One Tailed Tests Are Directional
2. One-tailed tests versus two-tailed tests
If you predict that trick-or-treaters in the mirror absent condition will take more candy than those in the mirror present condition, then use a one-tailed test
If you predict that trick-or-treaters in the mirror absent condition will take less candy than those in the mirror present condition, then use … a one-tailed test
Either way, you have a directional hypothesis, thus you use a one-tailed t-Test.
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Two Tailed Tests are Non-Directional
2. One-tailed tests versus two-tailed tests
For a general outcome (Group A will simply differ from Group B), the two-tailed test is best.
If you predict trick-or-treaters in the mirror absent condition will take a different amount of candy than those in the mirror present condition, then use a two-tailed test. Thus it could be higher OR lower
If you have a non-directional hypothesis, use a two tailed t-Test.
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One Tailed vs. Two Tailed
2. One-tailed tests versus two-tailed tests
It is easier to find significance using a one-tailed t-Test
Check yourself in Appendix B.2. If df = 1 (the first row), for a one-tailed test to be significant at the .05 level, you only need to have a t value above 6.314.
For a two-tailed test to be significant at the same .05 level, your t-value needs to be above 12.706!
That’s quite a high critical t value to overcome
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Pop-Quiz 3: Quiz Yourself
What t value for a one-tailed t-Test do you need to overcome if your df is 55 and your risk is p < .05?
A). 1.297
B). 1.673
C). 2.396
D). 1.673
E). 2.004
Answer 3: B
What t value for a one-tailed t-Test do you need to overcome if your df is 55 and your risk is p < .05?
A). 1.297
B). 1.673
C). 2.396
D). 1.673
E). 2.004
Pop-Quiz 4: Quiz Yourself
What t value for a two-tailed t-Test do you need to overcome if your df is 55 and your risk is p < .05?
A). 1.297
B). 1.673
C). 2.396
D). 1.673
E). 2.004
Answer 4: E
What t value for a two-tailed t-Test do you need to overcome if your df is 55 and your risk is p < .05?
A). 1.297
B). 1.673
C). 2.396
D). 1.673
E). 2.004
Why Would You Do A Two Tailed Test?
So why would you ever use a two-tailed t-Test if it is harder to find significance?
The next few slides go on a bit of a tangent about why you would use a one tailed vs a two tailed test.
Pay close attention, these concepts are important!
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More About Two Tailed Tests
A two-tailed test splits up the 5% error. Half is at the bottom of the normal curve (.025) while the other half is at the top (.025)
Just like we did with the one sample z-tests!
This is because your hypothesis is looking for a significant effect in both directions (that either group 1 has a higher mean OR that group 1 as a lower mean).
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The Normal Curve (Again)
This curve represents all the possible t-values (outcomes of our study).
t-values at the far right represent outcomes in which the mean for group 1 was higher than group 2
t-values at the far left represent outcomes in which the mean for group 1 was lower than group 2
With a two-tailed test, our hypothesis is that it could go either way!
Accepting The Null (two-tailed)
Null hypothesis: The groups do not differ, Ho: µ1 = µ2
Alternative hypothesis: The groups differ, H1:
if your t-value is here, you accept the null hypothesis
t-values of this size happen too frequently (95% of the time), it is not significant. The groups are equal (not different)
Accepting The Alternative (two-tailed)
Null hypothesis: The groups do not differ, Ho: µ1 = µ2
Alternative hypothesis: The groups differ, H1:
if your t-value is here, you accept the null hypothesis
if your t value is here or here, you accept the alternative hypothesis.
t-values of this size happen infrequently, it is probably not due to chance. The groups are different.
More About One-Tailed Tests
The one-tailed critical value is easier to overcome. All of that 5% error is on one side of the curve. Thus you need to be sure that the outcome would occur only 5 times out of a 100 by chance. No splitting percentages here!
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Accepting The Null (One-Tailed)
Null hypothesis: The groups do not differ or they differ in the opposite of the expected direction , Ho: µ1 µ2
Alternative hypothesis: The groups differ only in the expected direction, H1:
if your t-value is here, you accept the null hypothesis
t-values of this size happen too frequently (95% of the time), it is not significant. The groups are equal (not different), or they differ in the wrong direction.
Accepting The Null (One-Tailed)
Null hypothesis: The groups do not differ or differ in the opposite of the expected direction , Ho: µ1 µ2
Alternative hypothesis: The groups differ only in the expected direction, H1:
NOTE: Even if your t value is down here, in the opposite of the expected direction, this is still part of the null hypothesis (it is not the expected effect) Ho: µ1 µ2
if your t-value is here, you accept the null hypothesis
Accepting The Alternative (one-tailed)
Null hypothesis: The groups do not differ or differ in the opposite of the expected direction , Ho: µ1 µ2
Alternative hypothesis: The groups differ only in the expected direction, H1:
if your t-value is here, you accept the null hypothesis
if your t value is here, you accept the alternative hypothesis.
t-values of this size happen infrequently, it is probably not due to chance. Group 1 has a higher mean than Group 2
Summary: One Vs Two Tailed Tests
If you’re not sure which direction your results could go, you should do a two-tailed test.
If you do a one-tailed test, and you get the opposite of your expected findings, then you’d be unable to accept the alternative hypothesis! You’d have to accept the null.
Even if it’s harder to find a significant effect with a two-tailed test, it is worth doing if you think there is a chance your results might go either way.
Most researchers stick with a two-tailed test.
Degrees of Freedom
Wow, that was a lot…where were we again? Oh right, finding the critical value. We have the first two pieces of info we need to find the critical value, now we just need:
3. df (degrees of freedom)?
The third element in our Appendix table is the degrees of freedom, or the df. These are based on sample size for each group, and use the formula …
For our Halloween candy example, this is easy
df = (n1 – 1) + (n2 – 1)
df = (20 – 1) + (20 – 1) = 19 + 19 = 38
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Warning About The T-Table
Warning:
Sometimes you will see the actual df in the table, but not all values are listed there to save space.
Our Halloween study df of 38 is not listed in the t-test table, so we can look at either the df = 35 row or we can look at the df = 40 row.
Just go with the df that is closest (in our case, 38 is closer to 40 than 35),
As you see, the .05 critical values for df 35 (2.03) and df 40 (2.021) are really, really close, so there’s not much of a difference!
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Back to Step 5 (of 8)
Step Five: Determine The Value Needed To Reject The Null
Remember, we needed 3 pieces of information to find the critical value in the t-test table:
1. Your desired p values, that’s .05!
2. Whether you’re doing a One-Tailed test or a Two-Tailed
We’re doing a two-tailed test, based on our null and alternative hypotheses
3. df = 38
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Finding our Critical Value:
1. desired p value is .05!
2. Two-Tailed
3. df = 38 (remember, 38 isn’t listed here so we’ll go with 40 because it’s closest)
Our critical value is 2.021
Step 6 (of 8)
Step Six: Compare The Obtained And Critical Values
Here, our critical value is 2.021. Our obtained value is 5.43!
Our obtained value exceeds our critical value.
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Steps 7 & 8 (of 8)
Step Seven and Eight: Make A Decision
As we saw in step six, our obtained value of 5.43 exceeds the critical value of 2.03.
Thus we REJECT the null hypothesis and conclude that the mirror present and mirror absent groups differ significantly
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Pop-Quiz 5: Quiz Yourself
If we ran a one-tailed t-Test with two groups of 21 participants each and found a t obtained value of 1.75 (p < .05 level), what does this tell us?
A). We should retain the null hypothesis
B). We should reject the null hypothesis
C). There isn’t enough information here to render a conclusion
Answer 5: B
If we ran a one-tailed t-Test with two groups of 21 participants each and found a t obtained value of 1.75 (p < .05 level), what does this tell us?
B). We should reject the null hypothesis
The groups did differ
Your df is 40 (21 – 1) + (21 – 1) = 40
The critical t value is 1.68 at the p < .05 level
Your obtained value of 1.75 is high enough to overcome the critical t value of 1.68, thus you reject the null hypothesis
Pop-Quiz 6: Quiz Yourself
Let’s say you have a sample of 60 participants divided into two groups of 30. What df would you use when checking your critical value in a t-Test table?
A). 30
B). 48
C). 58
D). 60
E). 68
Answer 6: C
Let’s say you have a sample of 60 participants divided into two groups of 30. What df would you use when checking your critical value in a t-Test table?
A). 30
B). 48
C). 58 (n – 1) + (n – 1) = (30 – 1) + (30 – 1) = 29 + 29 = 58
D). 60
E). 68
Pop-Quiz 7: Quiz Yourself
There is no df of 58 in your t-Test table. Which df(s) could you use?
A). 50 or 60
B). 55 or 60
C). 60 or 65
D). 45 or 65
E). None of the above
Answer 7: B
There is no df of 58 in your t-Test table. Which df(s) could you use?
A). 50 or 60
B). 55 or 60
C). 60 or 65
D). 45 or 65
E). None of the above
Pop-Quiz 8: Quiz Yourself
For a two-tailed t-Test at a p < .05 level, what is the difference between a df of 55 and a df of 60?
A). 1.00
B). 0.10
C). 0.001
D). 0.002
E). 0.003
Answer 8: E
For a two-tailed t-Test at a p < .05 level, what is the difference between a df of 55 and a df of 60?
A). 1.00
B). 0.10
C). 0.001
D). 0.002
E). 0.003 df 60 = 2.001 df 55 = 2.004 2.004 – 2.001 = 0.003
So, What Now?
Sorry, still not done with our Halloween study! We know the groups differ, but we need to go back and look at the means to really understand HOW the groups differ.
To see what the direction of the effect is, we can look at the means…
Group #1: Mirror Present M = 4.00
Group #2: Mirror Absent M = 8.00
So, those without the mirror present took significantly MORE candy than those with the mirror.
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Writing Up The T-Test
Now we are nearly done. Our last step is to write this up as we would see it in a journal results section.
We’ll write up the t-test like this:
t(38) = 5.43, p < .05
Note, that p < .05 means it was significant. If it were not significant, it would say p > .05!
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Full Write-Up
Here’s the full write up for the study. It needs to include means, standard deviations, t-test, df, and p values, as well as a brief explanation of the findings.
“We ran an independent samples t-Test with mirror condition as our independent variable (mirror present versus mirror absent) and the amount of candy trick-or-treaters took as the dependent variable. The groups differed significantly, t(38) = 5.43, p < .001. Trick-or-treaters took more candy in the mirror absent condition (M = 8.00, SD = 1.52) than trick-or-treaters in the mirror present condition (M = 4.00, SD = 2.92). Apparently, seeing themselves in a mirror can lead participants to be less greedy!”
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Part Four
Special Effects: Are Those Differences For Real?
Are Those Differences Real? Effect Size
Special Effects: Are Those Differences Real?
So we found significant differences in our Halloween candy study, but are those differences meaningful?
Effect size is a measure of just how different two groups are from one another—that is, it is a measure of the magnitude of the treatment
The effect size does not rely on the sample size, so we can compare effect sizes between different studies!
So what is the magnitude of the our effect in our Halloween candy example? …
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Effect Size Formula
Here is our formula for Effect Size (ES), also known as Cohen’s d.
d= Effect Size (Cohn’s d)
= the mean of Group 1
= the mean of Group 2
= the variance of Group 1
= the variance of Group 2
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Effect Size for The Halloween Data
Here is the calculation of the effect size for our example:
= = 1.72
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Interpreting The Effect Size
So our effect size is 1.72. Is that high? Consider the guidelines that Jacob Cohen came up with to assess effect sizes:
1. A small effect size ranges from 0.00 to .20
2. A medium effect size ranges from .20 to .50
3. A large effect size is any value over .50
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Full Write up With Effect SIze
We can include the effect size in our write up:
“We ran an independent samples t-Test with mirror condition as our independent variable (mirror present versus mirror absent) and the amount of candy trick-or-treaters took as the dependent variable. The groups differed significantly, t(38) = 5.43, p < .05, d = 1.72. Trick-or-treaters took more candy in the mirror absent condition (M = 8.00, SD = 1.52) than trick-or-treaters in the mirror present condition (M = 4.00, SD = 2.92). Apparently, seeing themselves in a mirror can lead participants to be less greedy!”
The “d” is used to denote Cohen’s d!
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Tip: Use An Effect Size Calculator
A Very Cool Effect Size Calculator
As your book notes, we can get the effect size a lot easier by plugging in the values in an effect size calculator.
There are a lot of them available online. You can just google “effect size calculator” or go to this address:
Insert your mean and SD for both conditions, and then click “compute” and Cohen’s d will pop up!
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Why Use Effect Size?
Why do we even look at effect size?
As mentioned earlier, the effect size let’s us assess the overall magnitude of the difference. It’s not always enough to say two groups differ – we want to know if that difference is big!
Cohen's d is a measure of effect size. Simply put, it indicates the amount of difference between two groups.
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Reason 1: Effect Size
We use Cohen’s d for two specific reasons:
1. It is used as a counterpoint to significance tests were it now indicates how big or small that significant difference is. This difference can then be compared to Cohen's estimates of what is typical of a small, medium, or large effect.
As an example, suppose we found a difference in self-esteem in an experimental group compared to a control group. A Cohen's d of .50 suggest a big difference between the two conditions! Thus the difference is big!
If our groups were significantly different but Cohen’s d was really small (like 0.08), we could say that the groups differed significantly, but the difference was small.
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Reason 2: Effect SIze
2. We also use d to provide a common metric on which we can compare effects for meta-analysis (or other analyses) when outcome variables are measured on different scales.
That is, we can compare multiple studies using Cohen’s d, even if the different studies use different variables.
Because Cohen’s d is standardized, it allows us to make cross-study comparisons!
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Review: Calculating Variance and SD
Variance Calculations
The last several slides of this presentation go through the variance calculations for the two groups of data we had in our example.
Recall that I gave you these values before, but this is a reminder of how to calculate the variance.
This is just meant as a refresher, if you wanted to brush up on what we covered from Ch 3 (Salkind).
Computing The Variance
We must first compute the mean and the variance
The variance you’re also familiar with calculating!
Do you recall our variance formula? If not, here it is!
Since we have so many participants, we’ll need a few slides. Let’s start with the mirror present condition (two slides)…
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| Slide #1 (Condition #1 – Mirror Present – Participants 1 to 10 only) | |||||
| Subject # | X | (X-X) | (X-X)2 | ||
| S1 | 5 | 1 | 1 | ||
| S2 | 4 | 0 | 0 | ||
| S3 | 5 | 1 | 1 | ||
| S4 | 4 | 0 | 0 | ||
| S5 | 3 | -1 | 1 | ||
| S6 | 3 | -1 | 1 | ||
| S7 | 4 | 0 | 0 | ||
| S8 | 7 | 3 | 9 | ||
| S9 | 3 | -1 | 1 | ||
| S10 | 2 | -2 | 4 | ||
| Slide #2 (Condition #1 – Mirror Present – Participants 11 to 20 only) | |||||
| Subject # | X | (X-X) | (X-X)2 | ||
| S11 | 2 | -2 | 4 | ||
| S12 | 3 | -1 | 1 | ||
| S13 | 8 | 4 | 16 | ||
| S14 | 5 | 1 | 1 | ||
| S15 | 5 | 1 | 1 | ||
| S16 | 3 | -1 | 1 | ||
| S17 | 3 | -1 | 1 | ||
| S18 | 4 | 0 | 0 | ||
| S19 | 3 | -1 | 1 | ||
| S20 | 4 | 0 | 0 | ||
| Mean ( X ) for Condition # 1 (all 20 participants) was 80/20 = 4 |
Computing The Variance
Computing The Test Statistic – The Independent Samples t-Test
Step Four: Compute The Test Statistic Value (Condition #1)
Variance Continued
Now we just plug in our numbers (n = 20, so n – 1 = 19)
Okay, so condition #1 is done. Now for condition #2
= 2.3157
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| Slide #1 (Condition #2 – Mirror Absent – Participants 1 to 10 only) | |||||
| Subject # | X | (X-X) | (X-X)2 | ||
| S1 | 7 | -1 | 1 | ||
| S2 | 8 | 0 | 0 | ||
| S3 | 12 | 4 | 16 | ||
| S4 | 11 | 3 | 9 | ||
| S5 | 8 | 0 | 0 | ||
| S6 | 7 | -1 | 1 | ||
| S7 | 5 | -3 | 9 | ||
| S8 | 6 | -2 | 4 | ||
| S9 | 8 | 0 | 0 | ||
| S10 | 9 | 1 | 1 | ||
| Slide #2 (Condition #2 – Mirror Absent – Participants 11 to 20 only) | |||||
| Subject # | X | (X-X) | (X-X)2 | ||
| S11 | 9 | 1 | 1 | ||
| S12 | 10 | 2 | 4 | ||
| S13 | 13 | 5 | 25 | ||
| S14 | 14 | 6 | 36 | ||
| S15 | 7 | -1 | 1 | ||
| S16 | 6 | -2 | 4 | ||
| S17 | 3 | -5 | 25 | ||
| S18 | 8 | 0 | 0 | ||
| S19 | 4 | -4 | 16 | ||
| S20 | 5 | -3 | 9 | ||
| Mean ( X ) for Condition # 1 (all 20 participants) was 160/20 = 8 |
Computing The Variance
Computing The Test Statistic – The Independent Samples t-Test
Step Four: Compute The Test Statistic Value (Condition #2)
Now, sum the (X – X)2 for all 20 condition #2 participants
1 + 0 + 16 + 9 + 0 + 1 + 9 + 4 + 0 + 1 + 1 + 4 + 25 + 36 + 1 + 4 + 25 + 0 + 16 + 9 = 162
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Computing The Variance
Computing The Test Statistic – The Independent Samples t-Test
Step Four: Compute The Test Statistic Value (Condition #2)
Variance Continued
Now we just plug in our numbers (n = 20, so n – 1 = 19)
Note: This is the variance for condition #2
Note: If we need to know the standard deviation, all we do is take the square root of 8.5263, or √8.5263 = 2.9199
= 8.5263
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