1. Topic 6: Proportions and Probabilities

2. Proportions and probabilities
15 out of 81 students surveyed were smokers, so the proportion of those who smoke is 15/81 = Suppose each student’s name of placed on a card. I randomly select a card, note if the person is a smoker, and then return the card and shuffle the deck. After repeating this a sufficiently large number of times, the long-run proportion of smokers is what we call the probability that a randomly selected student will be a smoker. Connection: The probability of randomly selecting an individual with a certain characteristic is equal to the proportion of the population that has the characteristic.

3. Contingency Table
A table that summarizes data for two categorical variables is called a contingency table.

4. Row and column proportions
Row proportions are computed using row totals, and column proportions using column totals.
None
Small
Big
Total
Spam
149/367 = 0.406
168/367 = 0.458
50/367 = 0.136
1.000
Not spam
400/3554 = 0.113
2657/3554 = 0.748
495/3554 = 0.139
549/3921 = 0.140
2827/3921 = 0.721
545/3921 = 0.139

5. Row and column proportions
Row proportions are computed using row totals, and column proportions using column totals.
None
Small
Big
Total
Spam
149/367 = 0.406
168/367 = 0.458
50/367 = 0.136
1.000
Not spam
400/3554 = 0.113
2657/3554 = 0.748
495/3554 = 0.139
549/3921 = 0.140
2827/3921 = 0.721
545/3921 = 0.139
Conditional probabilities
P(None | Spam) = 0.406
P(None | Not Spam) = 0.113
Marginal probability
P(None) = 0.140

6. Row and column proportions
Row proportions are computed using row totals, and column proportions using column totals.
None
Small
Big
Total
Spam
149/549 = 0.271
168/2827 = 0.059
50/545 = 0.092
367/3921 = 0.094
Not spam
400/549 = 0.729
2657/2827 = 0.941
495/545 = 0.908
3684/3921 = 0.906
1.000

7. Row and column proportions
Row proportions are computed using row totals, and column proportions using column totals.
None
Small
Big
Total
Spam
149/549 = 0.271
168/2827 = 0.059
50/545 = 0.092
367/3921 = 0.094
Not spam
400/549 = 0.729
2657/2827 = 0.941
495/545 = 0.908
3684/3921 = 0.906
1.000
Conditional probabilities
P(Spam | None) = 0.271
P(Spam | Small) = 0.059
P(Spam | Big) = 0.092
Marginal probability
P(Spam) = 0.094

8. Area of histograms
10% of the data (2 out of 20) is in the 240 to 280 range.

9. Area of histograms
10% of the data (2 out of 20) is in the 240 to 280 range.
10% of the total area of the histogram is in the 240 to 280 range.