Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

6.1 & 6.2 Significance and Basic Probability Is it Normal?  Heights of 5,000 randomly selected sunflowers…  YES--NORMAL  The percentage of Coke six packs that contain six cans…  100%--UNIFORM  Sums of the means of rolling 1000 dice…  YES--NORMAL  A specially designed circuit with only an output of 110 volts…  NO--UNIFORM  Percentage of students with an IQ higher than 68 when the mean IQ score is 100 and σ = 16…  97.5%--NORMAL

6.1 & 6.2 Significance and Basic Probability STATISTICAL SIGNIFICANCE A set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance. Example: A detective in Detroit finds that 25 out of 62 guns used in crimes during the past week were sold by the same gun shop. Is this significant? Definitely! This is significant because, it is more than likely that there are many gun shops in the Detroit area, if 25 out of 62 guns come from the same shop, in the same week… CMS?? It’s very unlikely to have occurred by chance. {CMS---Common Mathematical Sense}

6.1 & 6.2 Significance and Basic Probability  Quantifying Statistical Significance… What is probability? Probability is the likelihood of an event occurring in terms of its statistical significance…MORE MATHEMATICAL CONNECTIONS TO PROBABILITY WILL FOLLOW. How can we quantify statistical significance and CMS {common mathematical sense} ? If the probability of an observed difference occurring by chance is 0.05 (or 1 in 20) or less, the difference is statistically significant at the 0.05 level. (In other words if the difference between the actual and observable variable is less than 5% then it is unlikely to have occurred by chance at the 0.05 level.)

6.1 & 6.2 Significance and Basic Probability  Quantifying Statistical Significance… If the probability of an observed difference occurring by chance is 0.01 (or 1 in 100) or less, the difference is statistically significant at the 0.01 level. (In other words if the difference between the actual and observable variable is less than 1% then it is unlikely to have occurred by chance at the 0.01 level.) Remember ---there is no guarantee for statistical significance. Correlation never implies causation!

6.1 & 6.2 Significance and Basic Probability  Quantifying Statistical Significance…  EXAMPLE: In the test of the Salk polio vaccine, 33 of 200,000 children in the treatment group got polio, while 115 of the 200,000 in the control group got polio. Based on the 0.01 significance level, is the difference statistically significant. ANSWER: Based on the large sample, the percent of chance is relatively high, however, the difference in cases must be less than 0.01 and optimistically it is statistically significant. Recall all significance is dependent on significance level, when in doubt use CMS. {Common Mathematical Sense}

6.1 & 6.2 Significance and Basic Probability  Quantifying Statistical Significance…  EXAMPLE: In a test on herbal cold remedies, 30 treatment subjects out of 100 contracted a cold. Also, 32 control subjects out of 100 contracted a cold. Based on the 0.01 significance level, is the difference statistically significant. ANSWER: Based on the sample of 100, the percent of chance is relatively high, however, the difference in cases must be less than 0.01 and hence it this particular case it is not statistically significant. Recall all significance is dependent on significance level, when in doubt use CMS. {Common Mathematical Sense}

6.1 & 6.2 Significance and Basic Probability  QUESTION & ANSWER TIME … Is the difference between occurred and chance value significant? 1. In 100 coin tosses, you observe 30 tails. Significant. 2. In 100 rolls of a number cube, a 3 appears 16 times. Not Significant 3. In 100 rolls of a number cube, a 2 appears 28 times. Significant 4. The first 20 cars you see during a trip are convertibles. Significant 5. An 85% free throw shooter hits 24 out of 30 free throws. Not Significant

6.1 & 6.2 Significance and Basic Probability  QUESTION & ANSWER TIME … Is the difference between occurred and chance value significant? 6. It rains for 5 days in a row in Phoenix in August. Significant students who attended a study session averaged a 78.9% on an exam, while 40 students who did not attend a study session averaged a 77.3% on an exam. Not Significant 8. If we assume the mean body temperature is 98.6, and the probability of finding a sample mean of 98.2 is Significant 9. The first 20 cars you see during a trip are Turquoise. Significant 10. A baseball team with a win/loss average of wins 12 out of 20 games. Not Significant

6.1 & 6.2 Significance and Basic Probability PROBABILITY: (Theoretical Method) The probability of an event, expressed P(event), is always between 0 and 1 inclusive. A probability of 0 means that the event is impossible, and a probability of 1 means that the event is certain.  Outcome – basic result of an observation or measurement  Event – a collection of one or more outcomes having the same property of interest. 1. Check all outcomes are equally likely 2. Count the total number of possible outcomes 3. Count the number of ways the event of interest, A, can occur. Probability Of An Event P(A) = The Number Of Ways Event A Can Occur The Total Number Of Possible Outcomes

6.1 & 6.2 Significance and Basic Probability  COUNTING OUTCOMES… Suppose process A has a possible outcomes, process B has b possible outcomes, and process C has c possible outcomes. Assuming the outcomes of the processes do not affect each other, the number of different outcomes for all of the processes combined is a×b×c. {Tree diagram and counting principle} Possibilities of tossing three coins

6.1 & 6.2 Significance and Basic Probability PROBABILITY: (Relative Frequency Method) Empirical probability, relative frequency, or experimental probability, is the ratio of the number favorable outcomes to the total number of trials, not in a sample space but in an actual sequence of experiments. In a more general sense, empirical probability estimates probabilities from experience and observation.relative frequencyexperienceobservation 1. Repeat or observe a process many times and count the number of times the event of interest, A, occurs. 2. Estimate P(A) =

6.1 & 6.2 Significance and Basic Probability PROBABILITY: (Subjective Method) A subjective probability describes an individual's personal judgment about how likely a particular event is to occur. It is not based on any precise computation but is often a reasonable assessment by a knowledgeable person. Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to 1; a rare event has a subjective probability close to 0, a very common event has a subjective probability close to 1. A person's subjective probability of an event describes his/her degree of belief in the event Estimate P(A) = “percentage of belief based on experience” PRACTICE???

6.1 & 6.2 Significance and Basic Probability PROBABILITY: Approaches? Identify the probability method. The chance that you will get married in the next year. Subjective Method, this is dependent on how you feel now. The chance of passing away in an automobile accident is 1 in 7,000. Relative Frequency Method, based on past DOMV data. The chance of rolling an even number on a number cube is 0.5. Theoretical Method, based on theoretical possible outcomes

6.1 & 6.2 Significance and Basic Probability Complement of an event…  The complement of an event, A, expressed as, consists of all outcomes in which A does not occur. The probability of is given by Ex. In a grocery store the scanning system was successful 384 out of 419. What is the probability that the scanner will not work?

6.1 & 6.2 Significance and Basic Probability Probability Distribution…  Is a table or graph or formula that gives the probability of all events. It has two properties: 1. Each probability must be a number between 0 and 1 inclusive. 2. The probabilities must total to 1. (Area under normal curve is 100%) Probability

6.1 & 6.2 Significance and Basic Probability HOMEWORK: Pg 236 # 1-10, 12, 14, 17, 21, 23 Pg 247 # 1-21 omit #12, and #29, 40