Elite Skier Example

Suppose a sports psychologist wants to study the effects of using an imagery technique on the performance of elite skiers.

DEPENDENT VARIABLE. She decides the dependent variable will be the elapsed time down some famous course. (Elapsed time is how long it takes skiers to ski down under racing conditions.) The psychologist assumes that elapsed time data can be modeled by a normal distribution.

INDEPENDENT VARIABLE. The independent variable for this study is using (or not using) an imagery technique (IT). The imagery technique is based on imagining in vivid detail that one is completing a perfect run down the race course.

TWO GROUP STUDY. She plans to do a two group study. She will randomly divide a pool of world-class, elite skiers into two groups. The treatment group will receive her special imagery technique (IT). The control group will receive no special treatment.

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Imagery Technique Population

The researcher expects that the imagery technique will improve the performance of elite skiers. Her expectation is that those elite skiers in the group which receives the imagery technique (IT) training will have lower elapsed times down the course than those without the IT training. In other words, she thinks that there are actually two populations--the Imagery (Treatment) Population and the untreated Control Population.

The IT population is expected to have a normal distribution, shown here as a red normal curve. For the moment, let's assume that the imagery technique (IT) works. Let's suppose that the mu of the treatment population is 120.47 seconds.

HOMOGENEITY OF VARIANCE. As we pointed out in the Detect Difference and Double Sample lecture, statisticians commonly assume that the two populations (control population and treatment population) have the same variability. That is, the sigma's for the two populations are equal.

ONLY MU's CAN DIFFER. In general the Normal Distribution has only two parameters (mu and sigma). Therefore, because we have assumed the sigma's are equal, the only potential distinction between the treatment and control populations comes from their mu's.

Two Populations

In summary, the researcher hypothesizes that the IT treatment is effective in improving performance. In probability theory, this scientific hypothesis is modeled in terms of there being two populations--the IT Treatment Population and the untreated Control Populaltion.

The Treatment and Control populations are assumed to have the same sigma.

In the graphic, the red distribution is the model of the elapsed times of the IT Treatment population and the green distribution is the model of the elapsed times of the Control or regular population that did not receive imagery training

The participants in the two groups are similar; both the Control Group and the Imagery Group are elite, world class skiers; they have comparable race times. The researcher will compare the performance of skiers sampled from the Control Population to performance those sampled from the Imagery Population in order to determine whether the imagery technique affects performance.

Even though researchers make hypotheses about populations, they hardly ever are able to test an entire population. Instead they collect samples from relevant populations. Then the researcher examines the sample data and makes an inference about the populations. This scientific thinking strategy is very similar to your experience with the Detect Difference Game. So think about your experience with that game as we continue on in this discussion.

For instance, this researcher wants to know if the IT technique will help elite skier's performance. She can't test all elite skiers in the world so she takes a random sample of elite skiers and gives half of them the IT training and the other half no special training. Then she looks at the performance of these two groups in order to make an inference about the effectiveness of the imagery technique.

Another way of thinking about the researcher's question is "Do the sample groups come from two different populations or from one population?" If the IT treatment has an effect on skiers' elapsed time on the course then the two groups come from different populations. If the treatment has no effect then the two groups come from the same population.

Ho versus H1. We will discuss this in greater detail further along in this lecture, but for now we can point out that the null hypothesis is equivalent to saying that there is only one population and the alternative hypothesis is equivalent to saying that there are two populations.

Here are the two populations again. For this example we have made up that the IT population has a mean (mu) elapsed time of 120.47 seconds and the control group or regular population has a mean (mu) elapsed time of 122.76 seconds.

The size of the theoretical treatment effect is easily calculated. It is the difference between the mu of the treatment population and the mu of the control population. In our example the treatment effect size is -1.29 seconds.

We have worked with this idea of treatment effect in the Double Sample tool and the Detect Difference lecture so that you should have lots of experience with it.

Larger treatment effects are easier to detect than smaller ones so if the mu's are far apart it is easier to determine that there are two populations rather than just one. When the distributions get closer and there is more overlap between the curves, then it is more likely that you will need a statistical test like a t-test to determine whether there are two populations or only one. The levels of difficulty on the Detect Difference Game are based on treatment effect size.

The scientist proposes that the treatment is effective and so there are two separate populations - the treatment and the control populations. In other words the IV has an effect upon the DV and therefore there really are two populations -- those who have been trained and can get down the hill faster, and those who haven't been trained who get down the hill slightly more slowly.

The skeptic, of course, does not believe in this imagery stuff. So to the skeptic, it doesn't really matter if someone receives training or not, it's all the same. In other words, the skeptic believes that the treatment is not effective and therefore both groups are coming from the same population. That is, there is only one population.

Homogeniety of Variance Assumption

Just to repeat: Statisticians assume that the two populations have the same sigma. That's a standard assumption for t tests and all of these kinds of statistical tests. The variances are assumed to be equal, and so the only possible difference between the two populations is their mu. The real question is, given that the sigma's are the same, "Is there a difference between the mu's?"

NOTATION: We need general symbols that go beyond this example. So, for this example, let's call the treatment population Population 1. Population 1 has as its center, Mu 1. Let's also call the control population Population 1, with Mu 2 as its center.

NULL HYPOTHESIS: Ho, the null hypothesis, comes from the skeptical hypothesis. One general form of Ho states that "Mu 1 minus Mu 2 is equal to zero." If the two population mu's are equal then their difference is zero. [Another way to say this is that if the two mu's are the same, then there's just one population.]

ALTERNATIVE HYPOTHESIS. On the other hand, the alternative hypothesis, H1, states that "Mu 1 minus Mu 2 is less than zero." Remember we are measuring elapsed times so smaller values on the DV indicate better skiing performance. In other words, the treatment population will have lower scores than the control population. Therefore, there are probably two distributions. This scientific hypothesis is directional.

The statement "Mu 1 minus Mu 2 is less than zero" implies there are two populations. The logic is that if the two mu's are different, then there have to be two populations.

How do we decide if our two groups come from different populations or from the same population? We decide in the same way you decided when you played the Detect Difference Game.

REVIEW: The dependent variable is elapsed time; the independent variable is whether or not the skiers receive the imagery treatment. There are two groups of skiers; one group was trained and the other was not. The two groups both race down the same course. The researcher finds that, on the average, the elapsed time for the imagery group is less than the control group. The scientist predicted this result.

The scientist wonders if she is sampling from two populations (as H1 suggests) or from only one population (as Ho indicates)?

By now you have enough experience with our probability tools to know that if Ho is correct and if you take two samples from the same population, you're going to get chance differences between them. The whole PCH of chance idea is that these data, the imagery group having lower elapsed times, could be due to chance alone. Another way of saying this is that the skeptic thinks these data could result from two samples from the same population.

This is the whole point of all the experience you had playing the Detect Difference Game. The idea is to put you in the formal situation of the scientist, and in the Detect Difference game, you don't see the populations. You have to use the data to guess whether there are one or two populations. After you make your guess the Detect Difference game gives you feedback by showing you the populations.

Scientists never get that kind of feedback because these populations are only models. They don't really exist. It may be worth just going back and playing the Detect Difference Game again to help solidify these ideas.

SUMMARY: We have two samples of data. Did the data come from two populations, a red and a green one, or are they just two samples from the single green population that differ by chance? We can't see the two populations, all we see is the data.

That is the conceptual background for the t-test. Let's review another research study example.

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The skeptic will not be deterred by the fact that the data pattern seems to fit the scientific hypothesis. As a first critique, the skeptic will claim that the pattern of results happened by chance alone. After all, one of the two groups had to have higher scores, so the scientists were just lucky that the group they predicted would be higher came out above the control.

For example, suppose we divide all the students in any classroom into two groups randomly or, at least, irregularly. Suppose we put all those sitting in the odd numbered rows into one group and students sitting in the even numbered rows in another group. Then suppose we measure them on anything (GPA, height, the number of calories they ate yesterday).

We would expect that the means of the two groups by chance alone would differ. The means would not be equal right down to the fifth decimal point. The mean of one group would be higher than the mean of the other group.

For example, if we measure caloric intake no one would claim (without great mental gyrations) that, for example, people who sit in odd numbered rows eat more calories than people who sit in even numbered rows just because the mean calories eaten yesterday by the odd numbered group came higher than the mean of the even numbered group. The point is, in a very general way, it is plausible to argue that the results of any research study occurred by chance alone. This is the Plausible Competing Hypotheses (PCH) of chance. It is the first issue you must address in a discussion with a skeptic. Formally, evaluating the PCH of chance is called "Statistical Conclusion Validity." Inferential statistics such as t-tests evaluate the statistical conclusion validity of research.

Skeptical Hypothesis

The skeptic hypothesizes that the IV (psychotherapy treatment) is ineffective, meaning it neither improves or worsens mental health. Because the IV is ineffective there will be only chance differences between the two groups in the study - the control group and the treatment group. The skeptic assumes that these chance differences between the groups is strong evidence that the two groups are from the same population with one mu.

The skeptic would look at the graph of the results and say, "Yes there are differences, of course, but they are only chance differences."

The scientist hypothesizes that the IV (psychotherapy treatment) is effective, meaning it improves mental health. Because the IV is effective there will be a significant difference between the two groups in the study - the control group and the treatment group. The scientist assumes that the difference between the groups is strong evidence that the two groups are different populations with different mu's as opposed to two mu's that are from the same population and differing only by chance.

The scientist would look at the graph of the results and say, "Yes there are differences, which are due to the effect of the treatment."

From Scientific and Skeptical Hypotheses to Statistical Hypotheses

Next we need to translate our scientific hypotheses into statistical hypotheses. The skeptic's plausible competing hypothesis that any differences in our measurements of mental health (DV) between the two groups occurred only due to chance will be expressed in statistics as the null hypothesis (Ho). The scientific hypothesis that differences in mental health measurements between the two groups are caused by psychotherapy will be expressed in statistics as the alternative hypothesis (H1).

In terms of this example, the skeptic expects that the difference between the two populations mu's will be zero. Another way of expressing this is to state the it is EXPECTED that difference between the two sample means will be zero.

You'll see both these forms in different books and I just want you to realize that those forms are exactly the same, logically speaking. That's what the skeptical hypothesis says in essence. Remember that the skeptical hypothesis goes with Ho. So Ho will be stated in a parallel way.

Ho in Symbols

TWO COMMON FORMS: As you can see from the graphic, Ho can be expressed in terms of the population means (top formula) or in terms of sample means (bottom formula).

Note: In the bottom formula, the large E is technically the"Expectation Operator" in probability theory. But for the intuitive level we are approaching the material in this class, you can assume E means pretty much what the word"expects" means in ordinary English. Ho says that the skeptic "expects" (E) [Mean 1 minus Mean 2] to equal zero. Of course, the actual sample means vary a great deal by chance alone, the skeptic EXPECTS the difference between the means to be zero.

The scientific hypothesis says that there are treatment effects of psychotherapy. Consequently the treatment population mu is higher than the control population mu.

This can also be expressed at the in talking about the data means. The scientist expects Mean 1 (the mean of the Experimental Group which was given psychotherapy) to be greater than Mean 2 (the mean of the control group which was not given psychotherapy). Is the scientific hypothesis directional or non-directional? Answer this question for yourself before going on.

In symbols H1 can be expressed in terms of treatment populations as in the top formula.

Or H1 can be expressed using the "Expectation Operator." According to H1 we EXPECT the difference between the data means to be greater than zero. This is because M1 comes from the psychotherapy group which should have higher mental health than does the control group.

Is the scientific hypothesis directional or non-directional?

The scientific hypothesis is directionalbecause the scientist is expecting the mean of the psychotherapy group to show higher mental health scores than the mean of the control group. That is, we are predicting a direction.

We're NOT just saying that the two means will be different somehow. We are NOT saying that it doesn't matter which way the means differ. We ARE specifying that the psychotherapy group should do better than the control group. That is why the scientific hypothesis is directional.

As we've said, a directional scientific hypothesis translates into a one-tailed alternative hypothesis. H1, the alternative hypothesis, says that we expect Mean 1 to be greater than Mean 2. We call this a one-tailed alternative. That is, we will reject Ho in one tail of the distribution.

We've been over this a little bit in the lecture on Hypothesis Testing. But we haven't developed the idea of one-tailed versus two-tailed tests fully yet. So we'll come back to this idea a little further along in the lecture.

Next we have n1, this is the number of subjects in group one. Correspondingly, n2 is the number of subjects in group two. The number of subjects in groups one and two can be different; you can have a different number of subjects in the two groups. Next we have the variances of the two groups. The variance of group one is s1 squared . The variance of group one is s2 squared. [Remember we are using the true variance formula which is divided by n. There is another variance formula used to estimate population variances; that formula has (n -1) as a divisor. For people who are using that formula t would have to have slight modifications.] An idea which we've not discussed before is the degrees of freedom (df). Right now, just note what the "df" symbol means in English. We will work with it more later, but as a start you can note that in this case df = n1 + n2 - 2.

That finishes the definitions of the various parts of the formula.

Supplemental Information: [You will not need this information in this class. So read this section only if it is of interest to you. It is provided as a response to a frequently asked student question.] STUDENT QUESTION: "Does the expected value of the difference between the means always have to equal zero if Ho is true?"

ANSWER: It is logically possible that you would have a case and a hypothesis in which Ho is actually predicting some known difference between groups. Let's think of an example. Okay, let's say there's a known difference in reaction time between men and women; women have faster reaction times than men. Let's say that difference is known to be 10 milliseconds. So you could do some intervention with men, perhaps you could give a group of men six cups of Café Gourmet's strongest brew. A comparable group of women would be given six cups of decaf. Suppose the scientific hypothesis is that caffeine will decrease reaction time (so the men should be responding as fast or faster than the women). The skeptic says that caffeine has no effect on reaction time.

If the skeptic is right, then your group of men should still be responding 10 milliseconds slower than women. So Ho would be that we still expect the difference between the means to be 10 milliseconds. Therefore, in this example for the E( M1 - M2 given Ho) term you would put 10 instead of 0. This is because 10 milliseconds actually represents no change as a result of your coffee intervention. So, theoretically, this term could have a value other than zero. But in this class we won't use those kinds of examples.

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Calculating t

This screen presents the data from our hypothetical experiment. The numbers are a result of our dependent variable measurement operations and indicate mental health, with higher numbers corresponding to better mental health.

The formula for the t test for independent means is presented so we can use it in the context of the example. BEFORE you go onto the next screen, the most important activity for you at this point is to use the data to substitute into t formula, writing the substitution in your class note outline. The next screen will show the correct substitution but you will learn the most by being active and substituting into the formula on your own.

When you have made your substitution go onto the next screen and carefully check to see if you understand how to use the formula. A series of screens will follow, showing the steps of the arithmetic, including the calculated t value and the degrees of freedom. When you are comfortable with how to use the formula, go on to the next section.

All t tests are either one-tailed or two-tailed. The one-tailed tests have either an upper or a lower rejection region. This refers to whether you are rejecting in only one tail of the distribution (the right long tail, or the left long tail) or whether you can reject in either of the two tails of the distribution (see the top distribution to the left).

To set up rejection regions, one has to decide two things. (1) Is it one- or two-tailed? (2) If it is one-tailed, does it have an upper rejection region or a lower rejection region?

The way to determine this, either in science, on an exam, or in the homework, is to read the scientific hypothesis. If the hypothesis is non-directional, then it goes with a two-tailed t test. This means that the researcher can reject Ho if the effect is far enough from zero either on the negative side or the positive side of the distribution.

The second requirement in using the table of critical values of t is that you must know whether you have a one- or a two-tailed alternative hypothesis. In our Psychotherapy example our H1 is one-tailed because the scientific hypothesis is directional: We expect the group given psychotherapy to have higher mental health scores.

Alpha Level

The final requirement is that we must specify an alpha (a) level. This is a choice that you have. The tables will list several alpha levels, including .05, .025, .01 and others. You choose the level. By convention, in the social sciences you ought not choose an a level larger than .05. Any of the smaller alpha levels, such as .01 or .001 is acceptable. For this example, let's choose alpha = .05. [Remember the alpha is the probability that you are wrong when you reject Ho. By choosing alpha, you are choosing the probability that you're wrong.] We have calculated the df = 7, we know we have a one-tailed test, and we have chosen alpha = .05. Now we are ready to look up the critical value in the statistical tables.

Your table will not look very much like the one on the screen because the screen just shows a schematic. There's only a few simple points to using the table.

First, the critical values are the numbers in the body of the table. You just have to go down some number of rows and across so many columns and read the number you find at the intersection of the row and column you choose. It's about like looking up the mileage from Las Vegas to Salt Lake City on a road map while sitting in a coffee shop. Use the schematic on the screen to follow along with the instructions below.

Critical Value Divides the Distribution

Now we know the exact value of our critical value. So we can put critical t = 1.895 right below the line we've drawn to divide the range of t into two regions. We are going to talk in more detail about what this picture means in the section on statistical conclusion validity.

Reject Ho in a
One-Tailed Lower Test

Just to review, here is how to find the critical value on the table. There's only a few simple points to using the table.

First, the critical values are the numbers in the body of the table. You just have to go down some number of rows and across so many columns and read the number you find there.

The degrees of freedom run down the side of the table. Our degrees of freedom (five plus four minus two) equal seven; so we go down to the row with "7" in front of it. Our critical value will be somewhere on this row. Next we go across the top of the table based on alpha. But there's one complication. Alpha is different if you have a one- versus a two-tailed test. So the table has a distinction between one- and two-tailed tests. A common symbol for this is Q (one-tailed) versus 2Q (two-tailed). Since we have a one-tailed test, go across the top of the table using the A's in the Q row. When you get to .05 you will be above the right column.

Then you'll come down to where you meet the seven degrees of freedom row. There, at the intersection of the row and column, will be the critical value, 1.895.

This is the critical value of t that we need for our particular example. However in this example you need to remember that the value in the table is an absolute value. For a one-tailed upper test you should consider the table's value to be equal to + 1.895 but for a one-tailed lower test, you should consider the value to be - 1.895 (a negative value less than zero).

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To discuss a two-tailed test we have to change our example again. Now our group of psychotherapy researchers are testing a new, controversial, and powerful therapy technique but they are unsure if this treatment will be beneficial or detrimental to the client. Now the scientific research team is unsure which of the two means will be greater or less than the other. They are unsure and think it could go either way. So the alternative or scientific hypothesis is just that M1 (the mean of the treatment group) could be either higher or lower than M2 (the mean of the control group). The scientists are not predicting the direction anymore; what they are predicting is that the two means will be very different from one another.

Since the test is no longer directional, the researchers are not predicting that the t value will be higher than 0 or less than 0. All they are predicting is the t value will be different from zero to such a magnitude that the researchers are fairly confident that they can reject Ho.

Again we are going to use the same basic process to determine whether the obtained or calculated t value is of a sufficient magnitude that the researcher can reject the null hypothesis. First, one determines the degrees of freedom associated with the test. Second, one determines whether it is a one- or two- tailed test. In this particular example we have determined that a two-tailed test is most appropriate to the research question. Finally one selects the alpha level for the test.

Just to review, for this example the researcher has selected a two-tailed and nondirectional test.

Finding the Critical Values for Two-Tailed Tests in the Table

Just as before, the critical values are the numbers in the body of the table. You just have to go down some number of rows and across so many columns and read the number you find there. Use the schematic on the screen to follow along with the instructions below.

Just as before, the degrees of freedom run down the side of the table. Our degrees of freedom (five plus four minus two) equal seven; so we go down to the row with "7" in front of it. Our critical value will be somewhere on this row. Next we go across the top of the table based on alpha. But there's one complication. Alpha is different if you have a one- versus a two-tailed test. So the table has a distinction between one- and two-tailed tests.

Since we now have a two-tailed test, go across the top of the table using the alpha's in the two-tailed row. When you get to .05 you will be above the correct column. It is essential that you make sure you are in the 2Q row since the Q row will give you the value associated with a one-tailed test and the critical value will be lower.

Then you'll come down to where you meet the seven degrees of freedom row. There, at the intersection of the row and column, will be the critical value, 2.365. This is the critical value of t that we need for this particular example. In this example you need to remember that the value in the table is an absolute value. For a two-tailed test you should consider the table's value to be equal to + 2.365 OR -2.365. If the researcher gets a calculated t value of less than - 2.365 or greater than a + 2.365, she can reject Ho with some assurance that she is correctly doing so.

Here is a picture that shows how the rejection regions fall on either side of zero. If the researcher gets a calculated t value of less than - 2.365 or greater than a + 2.365, she can reject Ho.

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