1. Module 15
Math

2. Analyze the distribution of categorical data
Student
Math anxiety level
Student 1
High
Student 2
Low
Student 3
Moderate
Student 4
Student 5
What percentage of students fall into each category?
Which response is the most common?
Is there a pattern in the response?
There are 1200 students in the study…this is only a sample. Can you answer these questions?

3. Analyze the distribution of categorical data
Category
Count
Proportion
Percentage
Low
110
110/1200
9.2%
Moderate
235
235/1200
19.6%
High
855
855/1200
71.3%
We can answer some questions but not all.

4. Analyze the distribution of categorical data
Low
Moderate
High
Row Totals
Female 37
163
560
760
Male 73 72
295
440
Column Totals
110
235
855
1200
We still have trouble comparing males versus females because there are more females then males. So what can we do?
Margins
Margins

5. Analyze the distribution of categorical data
Low
Moderate
High
Row Totals
Female
37/760 = 4.9%
163/760 = 21.4%
560/760 = 73.7%
760/760 = 100%
Male
73/440 = 16.6%
72/440 = 16.4%
295/440 = 67%
440/440 = 100%
Now we can compare males and females and look for a pattern.
Conditional Percentages

6. Calculate Marginal Percentages
18-34
35-54
55+
Total
iPhone
169
171
127
Android
214
189
100
Other
134
277
643

7. Calculate Marginal Percentages
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
How would you find the row (marginal) percentages?
What proportion of the total population own an iPhone?
What proportion of the total population own an Android?
What proportion of the total population own another type of phone?

8. Calculate conditional Percentages(probabilities)
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
How would you find the row (marginal) percentages?
What proportion of the total population own an iPhone?
What proportion of the total population own an Android?
What proportion of the total population own another type of phone?

9. Calculate conditional Percentages(probabilities)
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
P(the person owns an iPhone)?
P(the person is between 18-34)?
If we select a person between what is the probability that they own an Android? p 𝑎𝑛𝑑𝑟𝑜𝑖𝑑 18 −34 ?
𝑝 35−54 𝑜𝑡ℎ𝑒𝑟

10. Calculate Joint probabilities
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
If we select someone at random what is the probability that they are over 55 and own an iPhone?
𝑝 𝑜𝑣𝑒𝑟 55 𝑎𝑛𝑑 𝑜𝑤𝑛 𝑎𝑛 𝑖𝑃ℎ𝑜𝑛𝑒 = 127/2024
This is not conditional probability….we are not putting a condition on it.

11. Calculate probabilities
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
What is the probability that a randomly selected person will be in the category?
What type of probability is this?
= 31.5%

12. Calculate probabilities
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
P(Android and 18 – 34)?
What type of probability is this?
= 8.3%

13. Calculate probabilities
18-34
35-54
55+
Total
iPhone
169
171
127
467
Android
214
189
100
503
Other
134
277
643
1054
517
637
870
2024
18−34 𝑂𝑡ℎ𝑒𝑟 ?
What type of probability is this?
= 12.7%

14. Think Time How do you find marginal probabilities?
How do you find conditional probabilities?
How do you find joint probabilities?

15. Think time How are two-way tables different then:
Bar and circle graphs?
Scatterplots, histrograms, box plots, dot plots?
How are two-way tables similar too:
Scatterplots?

16. Risk vs. Probability They mean the same thing
Risk is when there is a negative outcome:
What is the probability you will win the lotto?
What is the risk of having an heart attack?
To calculate two risks we use the percent change formula
𝑁𝑒𝑤 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 −𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑟𝑖𝑠𝑘 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑟𝑖𝑠𝑘

17. Risk vs. probability
Does aspirin lower the risk of having a heart attack?
𝑝 ℎ𝑒𝑎𝑟𝑡 𝑎𝑡𝑡𝑎𝑐𝑘 𝑎𝑠𝑝𝑟𝑖𝑛
.013
.022
𝑝 ℎ𝑒𝑎𝑟𝑡 𝑎𝑡𝑡𝑎𝑐𝑘 𝑝𝑙𝑎𝑐𝑒𝑏𝑜
This means that taking asprin lowers the risk of a heart attack by 41%
.013 − =−0.41

18. Hypothetical 2 way table
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19. Hypothetical 2 way table
What is the probability that a dog chosen at random passed the course?
If the dog that was selected was a large dog, what is the probability that it passed the class?
200
250
450
400
150
550
600
400
1000

20. Hypothetical 2 way table
If the selected dog was small, what is the probability it passed the class?
Do you think dog size and whether or not a dog passes the class are related? Why or why not?
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150
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600
400
1000