Inference and Probability

1. Inference and Probability Module 10

2. Inference Overview • Recall that inference is using statistics from a sample to talk about a population. • We need some background in how we talk about populations and how we talk about samples.

3. Inference Overview • Describing a Population • It is common practice to use Greek letters when talking about a population. • We call the mean of a population . • We call the standard deviation of a population s and the variance . • When we are talking about percentages, we call the population proportion (the exception to using the Greek letter). • It is important to know that for a given population there is only one true mean and one true standard deviation and variance or one true proportion. • There is a special name for these values:

4. Inference Overview • Describing a Sample • It is common practice to use Roman letters when talking about a sample. • We call the mean of a sample . • We call the standard deviation of a sample s and the variance . • When we are talking about percentages, we call the sample proportion . • There are many different possible samples that could be taken from a given population. For each sample there may be a different mean, standard deviation, variance, or proportion. • There is a special name for these values:

5. Population Sample Mean: Standard Deviation: Proportion: Inference Overview • We use sample statistics to make inference about population parameters

6. Probability • The probability of an outcome is the proportion of times the outcome would occur if we repeated the procedure many times. • Examples • Coin: What is the probability of obtaining heads when flipping a coin? • A single die: What is the probability I will roll a four? • Two dice: What is the probability I will roll a four? • A jar of 30 red and 40 green jelly beans: What is the probability I will randomly select a red jelly bean? • Computer: In the past 20 times I used my computer, it crashed 4 times and didn’t crash 16 times. What is the probability my computer will crash next time I use it?

7. Probability • Independence: Two events are independent if the outcome of one does not affect or give an indication of the outcome of the other. Events Indepedent Dependent Flipping a coin twice Temperature on consecutive days 3 jelly beans: red, green, orange. Eat one. Eat another.

8. Probability • Independence: Two events are independent if the outcome of one does not affect or give an indication of the outcome of the other. Events Indepedent Dependent Randomly polling two individuals Comparing fertilizer yield for two adjacent field plots Rolling two dice

9. Probability • Definition: A sample space is a set of all the possible outcomes of a process. • Example: Coin • What is the sample space for flipping a coin 3 times?

10. Probability • Definition: An event is an outcome or set of outcomes of a process. • Example: Coin • What is one of the possible events for flipping a coin 3 times?

11. Probability Rules • Rule 1: The probability of any event is between 0 and 1 inclusive. • Pr(HTH) = • Rule 2: The probability of the whole sample space is 1. • Pr(rolling a 1 or 2 or 3 or 4 or 5 or 6) = • Rule 3: The probability of an event not occurring is 1 minus the probability of the event. • Pr(not rolling a 5) =

12. Probability Rules • Rule 4: If two events A and B have no outcomes in common, then Pr(A or B) = Pr(A) + Pr(B) • Pr(rolling a 1 or a 6) = • Rule 5: If two events A and B are independent, then Pr(A and B) = Pr(A)Pr(B) • Pr(rolling a 1 and then a 6) =