Monday, October 14 Statistical Inference and Probability

Population Sample You take a sample.

High Stakes Coin Flip

Let’s do an experiment.

The Coin Flip Experiment Question: Could the professor be a crook? Let’s do an experiment. Make assumptions about the professor. Determine sampling frame. Set up hypotheses based on assumptions. Collect data. Analyze data. Make decision whether he is or is not a crook.

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook.

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at.05

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at.05 Step 4. Decide on a sample, e.g., 6 flips.

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at.05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data.

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at.05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data. Step 6. Decide whether the data is more or less probable than . E.g., the probability of 6 consecutive wins based on the assumption in Step 1 is.016. (.5 x.5 x.5 x.5 x.5 x.5 =.016)

Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) =.5 Step 2. Set up hypotheses: H 0 : He is not a crook. H 1 : He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at.05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data. Step 6. Decide whether the data is more or less probable than . E.g., the probability of 6 consecutive wins based on the assumption in Step 1 is.016. (.5 x.5 x.5 x.5 x.5 x.5 =.016) Step 7. Based on this evidence, determine if the assumption that Hakuta is fair should be rejected or not.

What’s the probability of rolling a dice and getting 6?

Rolling a six (6) Six possible values (1,2,3,4,5,6) = 1/6 =.17

What’s the probability of rolling a dice and getting an even number?

Rolling an even (2, 4, 6) Six possible values (1,2,3,4,5,6) = 3/6 =.50

What the probability that your first (or next) child will be a girl?

It’s a girl! 2 possible outcomes (boy or girl) = 1/2 =.50

P(A) = Number of Examples of A Total Number of Sample Points

What the probability that your first (or next) child will be a girl and when she makes her first roll of dice, rolls an even number?

P(girl)  P(even).5 x.5 =.25

What the probability that your first (or next) child will be a girl or when that child makes his/her first roll of dice, rolls an even number?

What the probability that your first (or next) child will be a girl or when that child makes his/her first roll of dice, rolls an even number? List the possible outcomes. Girl, rolls even. Girl, rolls odd. Boy, rolls even. Boy, rolls odd.

What the probability that your first (or next) child will be a girl or when that child makes his/her first roll of dice, rolls an even number? List the possible outcomes.  Girl, rolls even.  Girl, rolls odd.  Boy, rolls even.  Boy, rolls odd.

What the probability that your first (or next) child will be a girl or when that child makes his/her first roll of dice, rolls an even number? List the possible outcomes.  Girl, rolls even.  Girl, rolls odd.  Boy, rolls even.  Boy, rolls odd..25

What the probability that your first (or next) child will be a girl or when that child makes his/her first roll of dice, rolls an even number? List the possible outcomes.  Girl, rolls even.  Girl, rolls odd.  Boy, rolls even.  Boy, rolls odd..25 .25 =.75

Addition Rule of Probabilities P(A  B) = P(A) + P(B) - P(A  B)

Addition Rule of Probabilities P(A  B) = P(A) + P(B) - P(A  B) P (A) = P(girl) =.5 P (B) = P (even) =.5 P (A  B) = P(A) x P(B) =.25 P(A  B) = =.75

Addition Rule of Probabilities P(A  B) = P(A) + P(B) - P(A  B) P (A) = P(girl) =.5 P (B) = P (even) =.5 P (A  B) = P(A) x P(B) =.25 P(A  B) = =.75

What is the probability of flipping 8 heads in a row?

.5 x.5 x.5 x.5 x.5 x.5 x.5 x.5 or.5 8 =.004

What is the probability of flipping 8 heads in a row?.5 x.5 x.5 x.5 x.5 x.5 x.5 x.5 or.5 8 =.004 Formalized as: The probability that A, which has probability P(A), will occur r times in r independent trials is: P(A) r

So, you decide to conduct a case study of 3 teachers, sampling randomly from a school district where 85% of the teacher are women. You end up with 3 male teachers. What do you conclude? P(males) three times = P(males) 3 =.15 3 =.003

So, you decide to conduct a case study of 3 teachers, sampling randomly from a school district where 85% of the teacher are women. You end up with 3 male teachers. What do you conclude? P(males) three times = P(males) 3 =.15 3 =.003 If you had ended up with 3 female teachers, would you have been surprised?