This assignment will allow you to practice conducting statistical analyses as a
researching psychologist. As you recall from class, prior to conducting a study
you need to ask a question that interests you. I provide an example below, but
you are instructed to create your own versions of interesting questions to ask
and to test using the supplied data.
You have been provided unique data sets so your statistics and answers
must be done on the data set that is assigned to you. A unique answer key
will be made for each data set and available to the TA to help them mark the
assignments. If your answers do not match the answer key, you will not receive
a full mark.
As discussed in class, hypothesis testing involves 4 steps.

1. Ask a question about the population and state the hypothesis
2. Use hypothesis to predict sample characteristics
3. Obtain sample, collect data, and perform statistical analyses
4. Compare result to prediction and make a decision
   3 Statistical tests you need to conduct
   You are required to come up with interesting questions/hypotheses for each
   of the tests below and perform the following statistical tests to address the
   questions. Please make sure that you are creating a unique question for each
   question. Get creative and make this activity fun for yourself. You can come
   up with silly and even unrealistic scenarios.
   1
   4 Questions
5. Perform a 1 sample t-test, 2-tailed, α = 0.05, and summarize your
   results 1
   (5 points)
6. Perform a repeated samples t-test, 2-tailed, α = 0.05, and summarize your results; Calculate Cohen’s d and report it in the conclusion (10
   points)
7. Perform a 2-sample independent samples t-test, 2-tailed, α = 0.05,
   and summarize your results; Calculate the 95% Confidence Intervals around
   the mean difference and report them in the conclusion (10 points)
8. Perform a 1-way ANOVA, 2-tailed, α = 0.05, on the 3 variables and
   summarize your results; show results for pairwise comparisons (10 points)
   For each question, show plot(s) of your data and write your conclusion/summary
   in APA format. If you are asked to report Cohen’s d or CIs, you may need to
   calculate these manually (or in R).
   1For this test, pick a meaningful population mean to which you will be comparing your
   scores
   2
   5 Example – 2-sample independent t-test, 2-tailed
   5.1 Question/hypothesis
   We would like to determine whether the resting heart rate (RHR) of students in
   PSY2116 is normal compared to the general population of University of Ottawa
   students. The reason that we are asking this question is because we want to
   know whether Dr. Konar is causing PSY2116 students to have heart conditions that may result in a different RHR compared to the population of all the
   University of Ottawa students.
   To conduct the study and to test our hypothesis, we take a sample of 20
   students from each population, n1 = 20 from PSY2116 and n2 = 20 from
   stopping students on campus at various locations and at different times to get a
   good representative random sample. We measure the RHR of each participant
   from both samples and record it (this is equivalent to the data stored under
   ‘V1’ for sample 1 and ‘V2’ for sample 2).
   Having measured the RHRs for all participants, we want to test our hypothesis that PSY2116 students have a different mean RHR compared to the general
   population because we suspect that Dr. Konar is causing students stress, which
   may present as a different RHR.
   When writing up my results, it helps to organize my thoughts using the
   sequence of steps we learned when conducting a hypothesis test.
   5.2 Hypothesis test
   Hence, my first step is to state my null and alternative hypotheses. The null
   hypothesis in this study is that Dr. Konar is not causing heart issues in his
   students and that the average RHR of PSY2116 students is not different from
   the general population of students at the University of Ottawa. The alternative
   hypothesis is that Dr. Konar is causing heart issues in PSY2116 students and
   thus their RHRs are different from the mean of the general population. Given
   that we are conducting a 2-tailed test, note the way I structured the formulaic
   version of the hypotheses:
   • H0: µ1 − µ2 = 0 or µ1 = µ2
   • H1: µ1 6= µ2
   You have the option of stating your hypotheses in words, as in the previous
   paragraph, or formulaically, as is seen right above this sentence.
   For the 2nd step, I want to figure out the criteria for accepting or rejecting my null hypothesis. For this, I will determine my tcritical using a table.
   Alternatively, I can rely on R’s built-in function that I will show below.
   For the 3rd step, we collect data (this was done for us already) and run the
   appropriate statistical analyses. Here, we calculate whether there is a significant
   difference between our sample data and the population, whether this difference
   3
   is meaningful (e.g., Cohen’s d), and use an interval estimate of the population
   mean (e.g., Confidence Intervals) instead of a point estimate (i.e., sample mean).
   For the 4th and final step, we make a decision after comparing tobserved to
   tcritical, and write our results in an APA format. Further, we report our effect
   size to supplement our t-test in order to determine whether the difference (if
   significant), was meaningful. Further, using Confidence Intervals, we can state
   how likely our mean is in relation to the population mean.
   6 Your data set
   Once you download the file ‘PSY2116-ClassData.zip’ from BrightSpace, you
   have to unzip it.
   • On Windows, note the directory where you are saving the file. In that
   directory, you can right-click on the file and choose Extract All; then pick
   a folder where you want the data to be placed. Pressing ‘Enter’ will just
   unzip in the same folder.
   • On OSX, note the directory where you are saving the file. If you doubleclick on the ‘PSY2116-ClassData.zip’ file, it should extract in the same
   directory in the Finder.
   • I haven’t done this on Linux in a long time, so if you cannot figure this
   out, please come see me and we’ll sort it out. I’m guessing if you are using
   Linux, this step is a non-issue.
   Find your student number among the files. This PSY2116 ∗.CSV
   file is your data set (your student # is in place of the *, obviously).
   Once you open your data file, it will include 3 columns of randomly generated
   numbers. The column names are ‘V1’, ‘V2’, and ‘V3’. There are 20 rows of data
   in each column. If you are asked to conduct an analysis on 1 sample, then use
   column 1 titled ‘V1’. If you are asked to conduct an analysis on 2 samples, then
   use columns 1 and 2 titled ‘V1’ and ‘V2’. Finally, when you are asked to do a
   1-way ANOVA, you will use all 3 columns for this analysis.
   The ‘PSY2116-ClassData.zip’ file also includes the file ‘PSY2116 1234567.csv’.
   I will be using the data from this file to demonstrate analyses in R. You can
   mimic how I do the analyses on this file and cater it to your data. Do NOT
   use the data from ‘PSY2116 1234567.csv’ as your own data. This file
   is for demonstrative purposes only.
   7 Getting data into R
   To import your data set into R, you have to follow a few steps that I will outline
   below.
   Assume that I saved the data file (‘PSY2116 1234567.csv’) on my Desktop.
   To import it into RStudio, open RStudio first. Then choose File, New File,
   4
   New R Script. All your work will be entered and stored here. Make sure to save
   this R script as something you’ll recognize, e.g., ‘PSY2116 Assignment.R’. Now
   enter the following commands into RStudio. Remember, to execute each line of
   code in RStudio. Windows: use Ctrl-Enter on the line of code that you would
   like to execute; OSX: use Command-Enter. Linux: hopefully you can figure this
   out; if not, talk to me and we’ll figure it out together.

Create a variable ‘file.location’ where you will specify the file’s

location:

on Windows, uncomment the following line (but make sure

to comment out the next line, which only works on OSX)

#

file.location = ‘c:/Users/yaro/Desktop/PSY2116_1234567.csv’

file.location = ‘~/Desktop/PSY2116_1234567DEMO.csv’ # <- on OSX

Please note that the user ‘yaro’ is specific to my computer ONLY.

Use your own user ID that you created in Windows in place of ‘yaro’

Now import the data from your *.CSV file into R.

Create an object ‘mydata’ where the data will be stored:

This function is telling RStudio that the file has a header and

data are separated by commas; all data will be stored in ‘mydata’

mydata = read.table(file.location, header = TRUE, sep = ‘,’)

View the data:

mydata

V1 V2 V3

1 41 25 20

2 43 28 31

3 57 19 38

4 43 31 38

5 51 27 35

6 33 27 34

7 62 28 30

8 52 31 23

9 55 28 35

10 47 29 40

11 39 28 27

12 43 18 31

13 50 31 21

14 43 34 24

15 55 28 31

16 53 31 29

5

17 46 38 30

18 45 28 30

19 51 29 21

20 51 26 28

Check your data, it should be a data.frame:

class(mydata)

[1] “data.frame”

Now spend some time familiarizing yourself with the data. You can do quick
descriptive stats and some plots to visualize the data.

Calculate mean of V1, the first column of data:

mean(mydata$V1)

[1] 48

same for V2:

mean(mydata$V2)

[1] 28.2

OR you can use a one-liner

with(mydata, mean(V1)); with(mydata, mean(V2))

[1] 48

[1] 28.2

Calculate the variance, standard deviation, etc

with(mydata, var(V1)); with(mydata, var(V2))

[1] 48.21053

[1] 19.43158

with(mydata, sd(V1)); with(mydata, sd(V2))

[1] 6.94338

[1] 4.408126

Plot boxplots of the data:

Look for whether the data are normally distributed, whether they

are skewed

with(mydata, boxplot(V1,V2))
6
1 2
20 30 40 50 60
7
Next, we want to do a 2-sample t-test on the imaginary data. For that I will
use the first 2 columns, where V1 will be the data for PSY2116 students and V2
will be data for the sample from the general University of Ottawa population.
So, for my dataset, this is how I would do a 2-sample t-test, 2-tailed:

The following lines of code show how to do a 2-sample t-test.

Make sure to select the following 4 lines of code,

then run the Cmd-Enter command:

with(mydata, # this is your data
t.test(V1, V2, # these are the 2 samples that you are comparing
alternative = ‘two.sided’, # specify that test is 2-tailed
var.equal = TRUE)) # assume that variances are equal

#

Two Sample t-test

#

data: V1 and V2

t = 10.766, df = 38, p-value = 4.215e-13

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

16.07704 23.52296

sample estimates:

mean of x mean of y

48.0 28.2

Pay attention to the t.test output because it tells you

whether to reject or fail to reject null. In this case,

the result is ‘reject null of no difference’. This is based on:

“alternative hypothesis: true difference in means is not equal to 0”

#

The CIs of -15 and -7 do NOT overlap 0, which indicates that the

mean difference is significant between V1 and V2.

You can confirm that the CIs are done correctly.

Below I show how to manually calculate the CIs around the

Sample Mean:

Recall that the CIs for 2-sample independent samples t-test are:

mu1 – mu2 = (M1 – M2) +/- t.critical*SE

the ‘t.critical*SE’ is an ‘error’ term that we can easily

calculate in R:

n1 = with(mydata, length(V1))
n2 = with(mydata, length(V2))
df = n1 + n2 – 2
8

Means of the data:

M1 = with(mydata, mean(V1))
M2 = with(mydata, mean(V2))

Since we have equal ‘n’, I will use the simple formula:

pooled.variance = with(mydata, (var(V1) + var(V2) ) /2 )
t.critical = qt(0.975, df) # or you can look this up in a table
SE = sqrt(pooled.variance/n1 + pooled.variance/n2)
error = t.critical * SE

Finally, the lower CI is

(M1-M2) – error

[1] 16.07704

Upper CI is

(M1-M2) + error

[1] 23.52296

To avoid this long process, you can simply use the result from

‘t.test’ function to save time.

Let’s recap what happened. In the above R code, ‘mydata’ is your data from
the .CSV file after you load it into R and save it into the variable ‘mydata’.
The next lines of code tell R to conduct a t-test on 2 variables that are inside
‘mydata’, V1 and V2. A quick way to test what names your columns have is to
run ‘names(mydata)’ in R. As a default, the t-test function calculates a 2-sided
test, so you can remove this line (or keep it for completion). You have to tell the
t-test function that the variances between your 2 data vectors are assumed to
be equal. As we have done in class, you can run a confidence interval manually
using a table in the book to look up tcritical. Alternatively, we can tell the t-test
function to do this for us. You just have to supply the level of confidence that
you want it to calculate.
For your report, I want you to report the tcritical. Instead of doing this
the hard way, by getting a textbook with t-tables, we can do this easily and
efficiently in R. However, it is useful to make sure your R value corresponds
to the one in the t-tables from a textbook. For example, for an α = 0.05 and
df = 10, the tcritical can be calculated with the following R code:

Calculating t-critical for alpha=0.05, 2-tailed, df=10:

alpha=0.05
df=10
qt(1-alpha/2, df)

[1] 2.228139

What will the t-critical be for alpha=0.05, 2-tailed test, for

9

your data?

HINT: substitute df=10 with the correct value for degrees of

freedom. What do you get?

For my data set, using an α = 0.05, df = n1 +n2 −2 = 20 + 20−2 = 38, the
tcritical = 2.024. Notice that the table in the book does NOT have df = 38, but
tcritical for df = 40 is 2.021, which is very close to my tcritical with a slightly
smaller df. Although it is standard procedure to use the smaller degrees of
freedom and not the larger one (like here), for the sake of this assignment we’ll
all choose the larger value. Using R, you are able to calculate the exact tcritical
for any df, even the ones that do not show up in the textbook tables.
Next, we want to calculate Cohen’s d.

To calculate Cohen’s d, we need to know the sample means

and pooled variance.

Sample means:

M1 = with(mydata, mean(V1))
M2 = with(mydata, mean(V2))

Degrees of freedom:

df1 = with(mydata, length(V1)-1)
df2 = with(mydata, length(V2)-1)

Variance:

variance1 = with(mydata, var(V1))
variance2 = with(mydata, var(V2))

Pooled Variance:

pooled.variance = (df1 * variance1 + df2 * variance2) / (df1 + df2)

Alternative way, using Sums of Squares:

This requires creating new columns where we calculate the

squared deviations from the mean, and then summing each column:

mydata$SquaredDeviation1 = with(mydata, (V1-mean(V1))^2)
mydata$SquaredDeviation2 = with(mydata, (V2-mean(V2))^2)
head(mydata) # shows 1st 6 rows of your data

V1 V2 V3 SquaredDeviation1 SquaredDeviation2

1 41 25 20 49 10.24

2 43 28 31 25 0.04

3 57 19 38 81 84.64

4 43 31 38 25 7.84

5 51 27 35 9 1.44

6 33 27 34 225 1.44

10
SS1 = with(mydata, sum(SquaredDeviation1))
SS2 = with(mydata, sum(SquaredDeviation2))
pooled.variance2 = (SS1 + SS2) / (df1 + df2)

Compare the 2 methods of calculating pooled variance:

pooled.variance

[1] 33.82105

pooled.variance2

[1] 33.82105

Cohen’s d:

(cohen.d = (M1-M2)/sqrt(pooled.variance))

[1] 3.404643

7.1 Summary/Conclusion
The final step of the process requires us to report our result in an APA format.
We found that the resting heart rate (RHR) of PSY2116 students (M = 48)
was significantly different from that of the general population of University of
Ottawa students (M = 28.2), t (38) = 10.77, p < 0.001, Cohen’s d = 3.4, 95%
CI [16.08, 23.52]. In other words, we found evidence that Dr. Konar is causing
PSY2116 students to have different heart rates than those found in the general
population of the University of Ottawa students. In fact, he is causing students
to have bradycardia, a lower heart rate of M = 48 compared to the general
student population of M = 28.2.