The Jaensch Effect

Question 1

Identify the Dependent and Independent Variables

The Independent variables are the Congruent and Incongruent list of words. The Dependent variables are the respective times it takes to read the lists of those words.

Question 2a

What is the hypotheses?

The Null hypothesis:

HO: There is no significant difference between the average time it will take to read the Congruent and Incongruent lists. Mathematically this is expressed as:

\[HO:\mu C - \mu I = 0\]

The Alternative hypotheses:

The Alternative hypotheses is that it will take more time to read the Incongruent list. Mathematically this is expressed as:

\[HO:\mu I - \mu C > 0\]

Question 2b

Which statistical test will distinguish the two hypothesis?

There are at least 3 tests to compare hypothysis like ours :the F-test , the Z-test, and the T-test.

The F-test compares variances of populations. This is appropriate when comparing the Consistancy of products of manufacturing machines. But we are interested in Causality not Consisitancy, did changing the color of the words cause longer reading times?

The Z-test compares a Sample v. a Population, e.g., comparing how well one statastics professor’s students perfomed compared to all of the students in the school who took that course.

There are 2 criteria to be met in order to perform the Z=test. 1. The Population Std Deviation must be known. 2. The sample size must be n>30.

We do not have the information to complete the formula. We fail to meet either criteria.

The T-test does not require either the Population Std Dev be known nor that the sample be greater than 30.

We will use a T-Test at 95% confidence level to compare the means of the Congruent and Incongruent. But should we use an Independent or Dependent T=Test? A single or two tailed test? An Independent sample would be the times it takes Machine A to pack a box v. Machine B to pack a box. A Paired or Dependent sample is when the same group is measured twice , before and after a treatment.In our machine exapmle , the times it took the same machine to pack a box before and afdter an change was made to the same machine. This describes our sample data . The same people were timed before and after a change was made. Now regarding a single or two tailed test our Ha is that not just that the mean times will be different but specifically that the “treatment group”, the Incongruent group, will have a greater mean time. So we will specify the parameter alternative = “less”. The alternative parameter makes the T-Test a single tail test . And the less specification asks the data if the mean of the first set of data, the Congruent, is significantly less than the mean in the subsequent, the Incongruent data.

# Question 3

What are some of the descriptive statistics?

Here the summary function is performed on the Stroop data set.

summary(stroopdata)
##    Congruent      Incongruent   
##  Min.   : 8.63   Min.   :15.69  
##  1st Qu.:11.90   1st Qu.:18.72  
##  Median :14.36   Median :21.02  
##  Mean   :14.05   Mean   :22.02  
##  3rd Qu.:16.20   3rd Qu.:24.05  
##  Max.   :22.33   Max.   :35.26

How big is the n ?

n<-nrow(stroopdata)
n
## [1] 24

What are the respective standard deviations?

s1<-sd(Congruent$Time)
s1
## [1] 3.559358
s2<-sd(Incongruent$Time)
s2
## [1] 4.797057

What are the values of the relelvant variables?

‘’’ n1<- length(Congruent$Time) # = 24

n2<- length(Congruent$Time) # = 24

x1<-mean(Congruent$Time) # = 14.26696

x2<-mean(Incongruent$Time) # = 22.33604

s1<-sd(Congruent$Time) # = 3.647708 Std Dev

s2<-sd(Incongruent$Time) # = 4.961341 Std Dev

df1<- n1-1 # = 24 Deg Free

df2<- n2-1 # = 24 Deg Free ‘’’ —

n1<- nrow(Congruent)        # = 24         number of observations          
n2<- nrow(Incongruent)      # = 24         number of observations            
x1<-mean(Congruent$Time)    # = 14.26696   Mean
x2<-mean(Incongruent$Time)  # = 22.33604   Mean
s1<-sd(Congruent$Time)      # = 3.559358   Std Dev
s2<-sd(Incongruent$Time)    # = 4.797057   Std Dev
df1<- n1-1                  # = 23         Deg Free
df2<- n2-1                  # = 23         Deg Free
## [1] 24
## [1] 24
## [1] 14.05113
## [1] 22.01592
## [1] 3.559358
## [1] 4.797057
## [1] 23
## [1] 23

Question 4

Illustrate the data

## [1] 18.03352

Here are the Congruent and Incongruent data undistinguished

Here are the Congruent and Incongruent distinguished.

The differences between Congruent and Incongruent are even more pronounced in a boxplot.

Observations on the visualizations.

The data whether combined or seperated is normally distributed.

Also, looking at the boxplot, it is worth noting that only the Incongruent data has any outliers but with that said more of its datapoints are captured in the middle 50% when compared to the Congruent.

Question 5

Perform statistical testing proceedures that include test statistic, p-vlaue, and test interpretation.

We will use R’s t.test() to compare our sample means. We chose the Paired T-Test becasue we are performing the 2 experiments on the same people. We have also set alt=“less” becasue we specifically are interested in finding out if the original times, Congruent, are less than the subsequent, Incongruent.

t.test(Congruent$Time, Incongruent$Time, paired=TRUE, alt="less" ,conf.level=0.95)
## 
##  Paired t-test
## 
## data:  Congruent$Time and Incongruent$Time
## t = -8.0207, df = 23, p-value = 0.00000002052
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf -6.262868
## sample estimates:
## mean of the differences 
##               -7.964792

The P-value is the probability that the Null hypothesis is True so a very low P-value indicates that there is a very low probabilty that the Null hypothesis is true. The criteria for assessing a P-value is the alpha which in our case is 0.05. If the P-value is equal to or below the alpha level we reject the Null hypothesis. Our P value at 0.00000002052 is well below our threshold of 0.05.

In the t-test we specify that pair = TRUE and that alternative = less becasue we are performing the test on the same people we took the baseline measurement of and alternative = less because specifically we want to know if the Congruent sample mean is less than the Incongruent. Or to put it the other way, have Times increased after the treatment? If we did not specify this we would be doing a two-tailed test and asking if our result was in the middle 95% Null region.

The critical value is 1.713872 and our t-value is well beyond that at 8.0207. This is consistant with our P-value.

abs(qt(0.05,23))
## [1] 1.713872

Lastly, the confidence interval,from -6.262868 and to negative infinity does not contain 0 so this too is consisitant with rejecting the Null.

Question 6

Conclusion Having tried the Stroop Test myself the results were expected. It is significantly harder to read the names of colors when the colors do not match.

Hypotheses to explain the reasons for the effect? Suggest an experiment to replicate the Stroop effect.

A Possible explanation for the Stroop effect is that we are asking our brains to read two things at once and then to ignore one of them.

Perhaps an experiment where people are shown arabic numbers in groups, e.g. 1 1 1 and 2 3 4 5 etc. and people are asked to state how many numbers are in each set?