11.1  Reviews
11.1  ReviewsIn this lesson you may need to use Minitab to construct frequency tables or twoway contingency tables. We'll start by reviewing these procedures. You will need to construct probability distribution plots for chisquare distributions. In earlier lessons we constructed probability distribution plots for z, t, and F distributions; the procedure is similar for a chisquare distribution. We will also review conditional probabilities and the term independence in this section.
11.1.1  Frequency Table
11.1.1  Frequency TableThe following example was first presented in Lesson 2.1.1.2.1.
It uses following data set (from College Board):
Minitab^{®} – Frequency Tables
To create a frequency table in Minitab:
 Open the Minitab file: sat_data.mpx
 Select Stat > Tables > Tally Individual Variables
 Double click the variable Region in the box on the left to insert the variable into the Variable box
 Under Statistics, check Counts and Percents
 Click OK
This should result in the following frequency table:
Tally
Region  Count  Percent 

ENC  5  9.80 
ESC  4  7.84 
MA  3  5.88 
MTN  8  15.69 
NE  6  11.76 
PAC  5  9.80 
SA  9  17.65 
WNC  7  13.73 
WSC  4  7.84 
N=  51 
11.1.2  TwoWay Contingency Table
11.1.2  TwoWay Contingency TableRecall from Lesson 2.1.2 that a twoway contingency table is a display of counts for two categorical variables in which the rows represented one variable and the columns represent a second variable. The starting point for analyzing the relationship between two categorical variables is to create a twoway contingency table. When one variable is obviously the explanatory variable, the convention is to use the explanatory variable to define the rows and the response variable to define the columns; this is not a hard and fast rule though.
Minitab^{®} – Constructing a TwoWay Contingency Table
 Open the data set: class_survey.mpx
 Select Stat > Tables > Cross Tabulation and Chisquare
 Select Raw data (categorical variable) from the drop down menu
 Double click the variable Smoke Cigarettes in the box on the left to insert the variable into the Rows box
 Double click the variable Biological Sex in the box on the left to insert the variable into the Columns box
 Click OK
This should result in the twoway table below:
Rows: Smokes Cigaretes  Columns: Biological Sex
Female  Male  All  

No  120  89  209 
Yes  7  10  17 
All  127  99  226 
Cell Contents: Count 
11.1.3  Probability Distribution Plots
11.1.3  Probability Distribution PlotsIn previous lessons you have constructed probabilities distribution plots for normal distributions, binomial distributions, and \(t\) distributions. This week you will use the same procedure to construct a probability distribution plot for the chisquare distribution.
Minitab^{®} – Constructing a Probability Distribution Plot
Chisquare tests of independence are always righttailed tests. Let's find the area of a chisquare distribution with 1 degree of freedom to the right of \(\chi^2 = 1.75\). In other words, we're looking up the \(p\) value associated with a chisquare test statistic of 1.75.
 In Minitab, select Graph > Probability Distribution Plot > View Probability
 Choose ChiSquare for the Distribution
 For Distribution select ChiSquare
 For Degrees of freedom enter 1
 Select A specified X value
 Select Right tail
 For X value enter 1.75
 Select OK and OK
This should result in the following output:
Example: Area to the Right of ChiSq = 6.25, df=3
Construct a chisquare distribution with 3 degrees of freedom to find the area to the right of a chisquare value of 6.25.
 In Minitab, select Graph > Probability Distribution Plot > View Probability
 Choose ChiSquare for the Distribution
 For Distribution select ChiSquare
 For Degrees of freedom enter 3
 Select A specified X value
 Select Right tail
 For X value enter 6.25
 Select OK and OK
The area to the right of 6.25 in the chisquare distribution with 3 degrees of freedom is 0.1001.
11.1.4  Conditional Probabilities and Independence
11.1.4  Conditional Probabilities and IndependenceIn Lesson 2 you were introduced to conditional probabilities and independent events. These definitions are reviewed below along with some examples.
Recall that if events A and B are independent then \(P(A) = P(A \mid B)\). In other words, whether or not event B occurs does not change the probability of event A occurring.
 Conditional Probability

The probability of one event occurring given that it is known that a second event has occurred. This is communicated using the symbol \(\mid\) which is read as "given."
For example, \(P(A\mid B)\) is read as "Probability of A given B."
 Independent Events
 Unrelated events. The outcome of one event does not impact the outcome of the other event.
Example: Queens & Hearts
If a card is randomly drawn from a standard 52card deck, the probability of the card being a queen is independent from the probability of the card being a heart. If I tell you that a randomly selected card is a queen, that does not change the likelihood of it being a heart, diamond, club, or spade.
Using a conditional probability to prove this:
\(P(Queen) = \dfrac{4}{52}=0.077\)
\(P(Queen \mid Heart) = \dfrac {1}{13} = 0.077\)
Example: Gender and Pass Rate
Data concerning two categorical variables can be displayed in a contingency table.
Pass  Did Not Pass  Total  
Men  6  9  15 
Women  10  15  25 
Total  16  24  40 
If gender and passing are independent, then the probability of passing will not change if a case's gender is known. This could be written as \(P(Pass) = P(Pass \mid Man)\).
\(P(Pass) = \dfrac{16}{40} = 0.4\)
\(P(Pass \mid Man) = \dfrac{6}{15}=0.4\)
In this sample, gender and passing are independent.