A Crash Course in Statistics at FIU - The t-Test (#2) - Fall 2022

So you took Stats I and Methods One at FIU and passed. But do you remember what you did, how you did it, and why you did it? If you need some basic statistic reminders for the t-Test, then this is the lecture for you! I am going to talk about a t-Test example in this document that corresponds to the example you saw in the Descriptive Statistics Crash Course (#1) where participants were asked to recall how much money they spent on textbook the prior semester.

While all participants were the eleventh person to write their textbook money recall on a survey the researcher gave them, the researcher actually completed the first ten slots, manipulating the amount prior "participants" recalled spending to be either high ($350 to $450) or low ($250 to $350). The researcher predicted that participants who saw high dollar amounts from the first ten participants would recall spending more money on textbooks themselves than participants who saw low dollar amounts from the first ten participants (demonstrating a conformity effect). The good news is that this mini-lecture will sum up the basics of the t-Test for you as we look at this study, though you can find additional information about the t-Test in your textbooks. On the final page are several questions based on this crash course. Answer these questions, and then go into your "Crash Course in Statistics - The t-Test Quiz #2" in your Canvas assessments menu and copy over your answer. Each Crash Course Quiz counts 5 points.

How, when, and why do a t-Test?

Before we get to the textbook money example, let me give you some basic information about the t-Test. The t-Test is used to compare two means to see if they differ significantly from one another. In this analysis, we need several pieces of information: the means for each group and the t-Test information itself.

Do you recall what a mean is? It is the average score. That is, you add up all of the scores and divide by the number of total scores to arrive at the average. Since a t-Test looks at two different conditions, this implies that you have two means: one for each condition. The means and standard deviation (as you saw in your first crash course quiz) are descriptive statistics. That is, they help describe the data.

The t-Test information itself is a test of inferential statistics. That is, we infer significant differences between the two groups. When writing it out, you will see a very common layout for the t-Test, something like: t(14) = 2.61, p = .021. The t tells you this is a t-Test. The 14 tells us our degrees of freedom (more on that in your lecture material). The 2.61 is the actual number for the t-Test. The p indicates whether it is significant (if it is less than .05, then it is significant).

We run a t-Test only under certain conditions.

First, our dependent variable (the variable we measure) must be continuous / scaled. That is, the DV has to be along a scale. For example, it can be an attitude ("On a scale of 1 to 9, how angry are you?"), a time frame ("How quickly did the salesperson help the customer on a scale of zero seconds to a thousand seconds?"), or a dollar amount ("How much do you recall spending on textbooks last semester?). We call these interval or ratio scales, and they allow us to run a t-Test. In comparison, we CANNOT run a t-Test on categorical data. That is, if we have a yes / no question ("Are you lonely: Yes or No") or a category based question ("What is your favorite food: hamburgers, pizza, salad, or tacos?"), then we cannot run a t-Test. These latter questions are based more on choice of option rather than an actual rating scale, and thus we cannot use a t-Test on them.

Second, we run a t-Test when we have two conditions to compare. That is, we compare the mean from Condition A to the mean from Condition B. If our t-Test is significant (p < .05), then we simply see which mean is higher. "The t-Test was significant, t(14) = 2.61, p = .021, with Condition A (mean = $337.50) higher than Condition B (mean = $293.75." Let's see how this looks using our "Cost of textbooks" example.

Data - Our "Money Spent on Textbooks" Study

Recall the basic set-up for our money spent on textbooks. Researchers ask participants to recall how much they spent on textbooks the prior semester, and each participant writes their answer on a survey sheet. In all conditions, the first ten answer slots are already filled in, presumably by other respondents. However, the researcher actually completed those ten slots, and manipulated the dollar amounts so that in in the High Dollar Condition, the first ten participants recalled amounts ranging from $350 to $450. In the Low Dollar Condition, amounts ranged from $250 to $350. Using social psychological principles based on conformity and informational social influence (e.g. participants relying on the behavior of other individuals when they lack a clear memory), the researcher predicts that those in the High Dollar Condition will recall spending more money on textbooks the prior semester than those in the Low Dollar Condition. Here, the independent variable is Dollar Condition (High versus Low) while the dependent variable is the amount of money participants recall spending on textbooks (in $). Figure 1 below represents the High Dollar Condition. Figure 2 represents the Low Dollar Condition.

Imagine we have eight real participants in the High Dollar Condition and eight real participants in the Low Dollar Condition (and no, we are not including the original dollar amounts on the survey passed out by the researcher, as those are not real participants!).

Consider the data:

Condition A (High Dollar Condition- in $)	Condition B (Low Dollar Condition- in $)
350	275
400	350
375	325
350	275
300	250
325	260
300	300
300	315
A = 2700	B = 2350
Mean = 337.50	Mean= 293.75
Median= 337.50	Median= 287.50
Mode= 300	Mode= 275
Standard Deviation= 37.80	Standard Deviation= 34.62

, or the symbol for Sigma, means "the sum of". Thus A is the sum of the scores for Condition A. That is, 350 + 400 + 375 + 350 + 300 + 325 + 300 + 300 = 2700. There are eight scores here, so we divide 2700 / 8 = 337.5, giving us our mean of $337.50 for Condition A (High Dollar Condition). We do the same thing for Condition B (Low Dollar Condition), giving us a mean of $293.75 (2350 / 8 = 293.75).

For the first part of our analysis, we compare the means. As you see, $293.75 in the Low Dollar Condition is less money than the $337.50 in High Dollar Condition. Eyeballing this, it looks like participants recall spending more money on books if they see other people reporting a high amount than if they see other people reporting a low amount. Thus the means for the High versus Low Dollar Conditions support our prediction, but we are not done yet. Just because the means seem to differ doesn't mean they do differ. To make that assessment, we run the t-Test and look at the p value to see if p < .05. We can do this by hand (like you did in Stats I and Methods One) or we can take the easy road and let SPSS calculate it for us. I am going to take the easy road, but keep in mind that we still have to interpret what SPSS tells us.

For the next section, I am going to open SPSS and run an independent samples t-Test. I'll use screenshots from SPSS as I go, but feel free to run these analyses yourself. Just set up your SPSS file like mine (I also included this SPSS file for you in Canvas if you prefer to use that. It is called "Crash Course Quiz #2 - Textbook Money (t-Test Practice)", but it is a short data set so I recommend setting up your own SPSS file using the values from the table above). I am just going to give you the basics here, but you can refer to other sources to figure out some of the info we get from the t-Test not covered in this lecture (like Levene's test, normality testing, etc.).

SPSS - Our Money Spent on Textbook Study

1. Click Analyze > Compare Means > Independent-Samples T Test... on the top menu as shown below.

You will be presented with the following:

1. Put the "Condition (1 = High, 2 = Low)" variable into the "Test Variable(s):" box and the "How much participants recall spending" variable into the "Grouping Variable:" box by highlighting the relevant variables and pressing the buttons. Note that SPSS uses different names for variables. It calls the dependent variable the "Test Variable(s)" and it calls the independent variable the "Grouping Variable". Just remember that our DV (the test variable) must be scaled in order to run this test (1 to 9, or 1 to 5, or even 0 to 100,000). The IV has to be categorical (High versus Low, Men versus Women, Old versus Young, Republican versus Democrat versus Independent, etc.).

1. You then need to define the groups (Condition). Press the button.

You will be presented with the following screen:

1. Enter "1" into the "Group 1:" box and enter "2" into the "Group 2:" box. Want to know why we do this? Well, remember that we labelled the High group as "1" and the Low group as "2"? That's why we use 1 and 2. NOTE: If you have more than 2 groups (e.g. a control condition with no money amount ("3") group, a moderate group with $300 to $400 ("4"), an extremely high group with $900 to $1,000 ("5") group, etc., you could type in "1" to "Group 1:" box and "3" to "Group 2:" box to compare the High condition and the No money condition. You could also compare the moderate money condition (4) to the extremely high condition (5) for a very interesting study! Since we only have two groups here, we will stick with 1 and 2

1. Press the button.

1. Then click okay (Ignore "Options" and "Bootstrap")

Output of the Independent t-Test in SPSS

You will be presented with two tables containing all the data generated by the Independent t-test procedure in SPSS.

Group Statistics (e.g. Descriptive Statistics)

The group statistics table provides useful descriptive statistics for the two groups you compared, including the mean and standard deviation. Remember, these "describe" the data for us, giving us the measure of center (e.g. the mean) and the measure of spread (e.g. the standard deviation).

As you can see, we have 8 participants in the High Dollar Condition and 8 participants in the Low Dollar Condition. The mean for High is $337.50 (SD = $37.80) and the mean for the Low is $293.75 (SD = $34.62). You might recall that these are the same means we saw in Crash Course #1 on Descriptive Statistics when we used the "Means" procedure. The nice thing about the t-Test is that it also gives us this information. Ignore the Std. Error Mean column, but we will need the rest of the information in our write up (below), so we'll come back to this table.

Independent Samples Test Table

This table provides the independent t-test results and Levene's Test for Equality of Variances.

We will talk about Levene's test later in the lecture, but I want to ignore it for now. Let me make the output a bit larger so we can interpret it properly. I'll remove several unneeded columns.

Above is the important information we will need for our t-Test (outlined in the red box). We will use the equal variances assumed row (it doesn't differ much from equal variances not assumed anyway, but I prefer to use the df that has a whole number). What we do now is write it up. For the write up, we need both of our tables - the descriptive table (with our means and standard deviations) as well as our independent samples t-Test table. Let me repost the descriptives table below (the "group statistics"). The important info is in the green box.

Reporting the output of the independent t-Test

We report the statistics in this format: t(degrees of freedom[df]) = t-value, p = significance level. In our case this would be: t(14) = 2.41, p = 0.03, and our means/SDs would be M = 337.50, SD = 37.80for the High Dollar Condition andM = 293.75, SD = 34.62for the Low Dollar Condition.We report the results as follows (including $ since we are dealing with money):

We ran an independent samples t-Test with Dollar Condition (High versus Low) as our independent variable and how much participants recalled spending (in $) as our dependent variable. The t-Test was significant, t(14) = 2.41, p = .03. Participants recalled spending more money on textbooks in the High Dollar Condition (M = $337.50, SD = $37.80) than in the Low Dollar Condition (M = $293.75, SD = $34.62).

Note that according to the 7^th Edition of the APA publication manual you should use exact p values. Here, p = .03 (rather than p < .05, which is the old but non-preferred way of writing up the p value). So p values like p = .34, p = .012, p = .001 are all formatted correctly while p < .05 is not. Only use the < sign if your statistical calculation is .000. That is, use p < .001 if SPSS or another statistical program (or your hand-based calculator!) shows p = .000 or less.

Of course, our t-Test might not be significant. In that case, we use a similar write-up but draw different conclusions:

We ran an independent samples t-Test with Dollar Condition (High versus Low) as our independent variable and how much participants recalled spending (in $) as our dependent variable. The t-Test was not significant, t(14) = 1.41, p = .235. Participants recalled spending similar amounts of money on textbooks in the High Dollar Condition (M = $337.50, SD = $37.80) and the Low Dollar Condition (M = $333.75, SD = $34.62).

That's it! Not too hard, right? Just remember the basics here: we use a t-Test to look at the differences between two means to see if the means differ significantly. We need two SPSS tables to make this assessment: the descriptive statistics tables ("group statistics") and the t-Test table ("Independent Samples t-Test").

Scroll down to the next page for the Crash Course Quiz #2 - the t-Test

Crash Course In Statistics - The t-Test - Quiz #2 (Deindividuation, Fall 2022)

Instructions: Have you ever sat in a movie theater and seen a short public service announcement (or PSA) for the Will Rogers Institute? These PSAs typically ask moviegoers to donate to the Will Roger's Foundation, a charitable organization that provides medical aide to those suffering from lung or pulmonary issue, as well as other rehabilitation and medical education needs. If you have seen these PSAs, did you donate after seeing them in a theater? Would the number of people in the theater impact your willingness to donate?

Research on deindividuation suggests that people tend to lose both their self-awareness and their sense of evaluation apprehension when they are in a crowd. For example, this may lead normally law-abiding people to riot and loot when part of a large angry mob since they cannot be individually identified, and it might decrease their willingness to be pro-social since they may find similar anonymity in a large crowd. So how might deindividuation apply when it comes to move theater patrons who are watching a charitable PSA for the Will Rogers Institute?

Imagine we run a study to assess the role of deindividuation in a movie theater. In a prior study we give all participants a free movie ticket for them and a friend to attend a movie on a specific date. What they don't know is that the study isn't over yet! In fact, we alter the nature of the crowd in the theater they go to, with some participants (and their friend) finding themselves the only two people in the theater ("Mostly Empty" condition) or in a theater that is mostly full of moviegoers ("Mostly Full" condition). Before the movie begins, all attendees watch the same PSA from the Will Rogers Institute, and the researchers pass out a questionnaire to all patrons regarding the Institute. One question asks them to rate how willing they are to donate $10 to the Will Rogers institute on a scale that ranges from 0 (I am not willing donate) to 10 (I am very willing to donate). All questionnaires are collected before the movie begins.

The authors think that participants will be more likely to donate when they are easily identifiable as an individual rather than when they are one person in a large crowd. Thus, the authors predict that participants willbe more willing to donate when they are in a "Mostly Empty" theater than when they are in a "Mostly Full" theater.

Using this study set-up, answer the questions below and then transfer those answers to your Crash Course in Statistics - The t-Test Quiz #2 in Canvas (1 point per question). IMPORTANT: The answer options on Canvas may not be in the same order you see them below, so make sure to copy over the CONTENT of the answer and not simply the answer letter (A, B, C, D, or E). Note: If you want to run these analyses yourself, look for the SPSS file called "#2 t-Test Crash Course Data Deindividuation Fall" in Canvas - not required, but definitely recommended!)

1). What is the independent variable in this study?

A. Whether the theater is Mostly Empty, Half Full, or Mostly Full

B. Whether the theater is Mostly Empty or Mostly Full

C. Ratings of how willing they are to donate money on a 0 (I am not willing to donate) to 10 (I am very willing to donate) scale

D. Ratings of how much money participants donate from $0 to $10

E. There is too little information in this study to determine the independent variable.

2). What is the dependent variable in this study?

A. Whether the theater is Mostly Empty, Half Full, or Mostly Full

B. Whether the theater is Mostly Empty or Mostly Full

C. Ratings of how willing they are to donate money on a 0 (I am not willing to donate) to 10 (I am very willing to donate) scale

D. Ratings of how much money participants donate from $0 to $10

E. There is too little information in this study to determine the dependent variable.

You run a t-Test on this data set and get the following SPSS output (note that I edited out some columns that you don't need for your write-up). Using this output, interpret the information.

3). Choose the correct means and standard deviations for the Mostly Empty and Mostly Full conditions. Make sure to round to two decimal places

A. The Mostly Empty condition has a mean of 6.48 and a standard deviation of 0.23 while the Mostly Full condition has a mean of 7.28 and standard deviation of 0.98.

B. The Mostly Empty condition has a mean of 0.98 and a standard deviation of 7.28 while the Mostly Full condition has a mean of 0.82 and standard deviation of 6.48.

C. The Mostly Empty condition has a mean of 25 and a standard deviation of 7.28 while the Mostly Full condition has a mean of 25 and standard deviation of 6.48.

D. The Mostly Empty condition has a mean of 7.28 and a standard deviation of 0.98 while the Mostly Full condition has a mean of 6.48 and standard deviation of 0.82.

E. The Mostly Empty condition has a mean of 3.13 and a standard deviation of 0.49 while the Mostly Full condition has a mean of 46.60 and standard deviation of 3.13.

4). Is the t-Test significant, and how would you write that out in APA format?

A. No, the t-Test is not significant, t(46.60) = 3.13, p = .30

B. No, the t-Test is not significant, t(48) = 0.49, p = .49

C.Yes, the t-Test is significant, t(48) = 3.13, p = .003

D.Yes, the t-Test is significant, t(46.60) = 3.13, p = .003

E. Yes, the t-Test is significant, t(48) = 0.49, p = .49

5). Finally, which of the following is the correct results as you would write them in an APA formatted results section.

A. We ran an independent samples t-Test with theater condition (Mostly Empty versus Mostly Full) as our independent variable and ratings of"How willing are you to donate?" as our dependent variable. The t-Test was not significant,t(46.60) = 3.13, p = .30. Participants were equally willing to donate in the Mostly Empty condition (M = 7.28, SD = 0.98) and the Mostly Full condition (M = 6.48, SD = 0.82).

B. We ran an independent samples t-Test with theater condition (Mostly Empty versus Mostly Full) as our independent variable and ratings of"How willing are you to donate?" as our dependent variable. The t-Test was not significant,t(48) = 0.49, p = .49. Participants were equally willing to donate in the Mostly Empty condition (M = 7.28, SD = 0.98) and the Mostly Full condition (M = 6.48, SD = 0.82).

C. We ran an independent samples t-Test with theater condition (Mostly Empty versus Mostly Full) as our independent variable and ratings of"How willing are you to donate?" as our dependent variable. The t-Test was significant,t(48) = 3.13, p = .003. Participants were less willing to donate in the Mostly Empty condition (M = 7.28, SD = 0.98) than in the Mostly Full condition (M = 6.48, SD = 0.82).

D. We ran an independent samples t-Test with theater condition (Mostly Empty versus Mostly Full) as our independent variable and ratings of"How willing are you to donate?" as our dependent variable. The t-Test was significant,t(48) = 3.13, p = .003. Participants were more willing to donate in the Mostly Empty condition (M = 7.28, SD = 0.98) than in the Mostly Full condition (M = 6.48, SD = 0.82).

E.We ran an independent samples t-Test with theater condition (Mostly Empty versus Mostly Full) as our independent variable and ratings of"How willing are you to donate?" as our dependent variable. The t-Test was significant,t(46.60) = 3.13, p = .003. Participants were more willing to donate in the Mostly Empty condition (M = 7.28, SD = 0.98) than in the Mostly Full condition (M = 6.48, SD = 0.82).

Thank you