Module 15 Math 075 2016

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?

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.

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

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

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

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?

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?

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 18-34 what is the probability that they own an Android? p 𝑎𝑛𝑑𝑟𝑜𝑖𝑑 18 −34 ? 𝑝 35−54 𝑜𝑡ℎ𝑒𝑟

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.

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 35-54 category? What type of probability is this? 637 2024 = 31.5%

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? 169 2024 = 8.3%

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? 134 1054 = 12.7%

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

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?

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

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 − .022 .013 =−0.41

Hypothetical 2 way table 200 250 450 400 150 550 600 400 1000

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

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? 200 250 450 400 150 550 600 400 1000