Chapter 5 & 6 Review. Producing Data Probability & Simulation

Transcription

1 Chapter 5 & 6 Review Producing Data Probability & Simulation

2 M&M s Given a bag of M&M s: What s my population? How can I take a simple random sample (SRS) from the bag? How could you introduce bias?

3 Population We are almost always interested in knowledge about a population. We would have little interest in samples if we could always ask everyone what they think about any particular issue that is, we would conduct a census. The reality is that we can t, so we need to get a sample that is representative of the population (I.e., shares characteristics of the population!)

4 Bias Sampling Why are these bad? Voluntary Response Sample Surveying people on customer satisfaction who come out of the DMV Convenience Sampling A survey on utilization of the cafeteria a CCA basketball game because it is convenient A survey about political views at a mall

5 Simple Random Sample

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7 Technology Tip Example 5.5 demonstrates the use of the random number table to select an SRS of size 5 from a population of size 30. To do the same thing on your calcuator, select MATH PRB randint(1,30) and press ENTER five times to get your sample. Ignore repeats. Or: randint(1,30,5) (works fine as long as there are no repeats).

8 Sampling Techniques Bad sampling techniques that do not produce good data: Convenience Sampling and Voluntary response sampling. Good sampling techniques that do produce good data: Probability sampling; it is a sample chosen by chance. We must know what samples are possible and what probability each possible sample has.

9 Types of bias Never say biased Name the bias! Response Bias (respondents lie, behavior of respondent or of the interviewer influences results, voluntary response bias) Wording of Questions Undercoverage Nonresponse

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11 Statistical Significance

12 Simulation Steps 1) State the problem or describe the random phenomenon. 2) State the assumptions. 3) Assign digits to represent outcomes. 4) Simulate many repetitions. 5) State your conclusions. Ex: Toss a coin 10 times. What s the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails?

13 Ex: Toss a coin 10 times. What s the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? Line 101 in Table B: Let: odds = heads; evens = tails If you repeat 25 times 23 of them have a run of 3 or more heads or tails => Estimated probability = 23/25 =.92

14 6.2 Probability Models Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run! Random is not the same as haphazard! It s a description of a kind of order that emerges in the long run. The idea of probability is empirical. It is based on observation rather than theorizing = you must observe trials in order to pin down a probability! The relative frequencies of random phenomena seem to settle down to fixed values in the long run. Ex: Coin tosses; relative frequency of heads is erratic in 2 or 10 tosses, but gets stable after several thousand tosses!

15 Exploring Randomness 1) You must have a long series of independent trials. 2) The idea of probability is empirical (need to observe real-world examples) 3) Computer simulations are useful (to get several thousand of trials in order to pin down probability)

16 Sample space for trails involving flipping a coin =? Sample space for rolling a die =? Probability distribution for flipping a coin =? Probability distribution for rolling a die =?

17 Event 1: Flipping a coin Event 2: Rolling a die 1) How many outcomes are there? List the sample space. Tree diagram: * Rule 2) Find the probability of flipping a head and rolling a 3. Find the probability of flipping a tail and rolling a 6. 3) # of outcomes?

18 Example: P(H and 3) = 1/2 * 1/6 = 1/12 P(T and 6) =

19 Sampling with replacement: If you draw from the original sample and put back whatever you draw out Sampling without replacement: If you draw from the original sample and do not put back whatever you drew out! EXAMPLE: 1) Find the probability of getting one ace, then another ace without replacement. 2) Find the probability of getting one ace, then another ace with replacement.

20 Disjoint/Complement Mutually exclusive No outcomes in common Can t happen simultaneously The probability that an event does not occur

21 Probability Rules: P(A U B) = P(A) + P(B) => A or B A union B is the set of all outcomes that are either in A or in B.

22 Venn Diagram: Union ( Or /Addition Rule) Find: 1) P(A U B) =getting an even number or a number greater than or equal to 5 or both 2) P(A or C) =getting an even number or a number less than or equal to 3 or both 3) P(B U C)=getting a number that is at most 3 or at least 5 or both.

23 The probability that BOTH events A and B occur A and B are the overlapping area common to both A and B Only for INDEPENDENT events Independent: knowing that one occurs does not change the probability that the other occurs

24 Venn Diagram: Intersection ( And /* Rule) Find: 1) P(A U B) =getting an even number that is at least 5 U 2) P(A and C) =getting an even number that is at most 3 3) P(B C)=getting a number that is at most 3 and at least 5. U

25 The Big Picture + Rule holds if A and B are disjoint/mutually exclusive * Rule holds if A and B are independent * Disjoint events cannot be independent! Mutual exclusivity implies that if event A happens, event B CANNOT happen.

26 Conditional probability: Pre-set condition ( given ) Find: 1) P(A given C) =getting an even number GIVEN that the number is at most 3. 2) P(A B) =getting an even number GIVEN that the number is at least 5.

27 If events A and B are not disjoint, they can occur simultaneously. Outcomes in common!

28 Let A = the woman chosen is Let B = the woman is married 1) P(A) 2) P(A and B) 3) P(B given A)

29 The probability we assign to an event if we know that some other event has occurred.

30 Example Seventy-five percent of people who purchase hair dryers are women. Of these women purchases of hair dryers, thirty percent are over 50 years old. What is the probability that a randomly selected hair dryer purchaser is a woman over 50 years old?