Probability in Statistics Chapter 6 Probability in Statistics

The Role of Probability in Statistics : 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 Toss of a fair coin 50-50 30-70 20-80 GED111/CDS111 Statistics in Modern Society

From Sample to Population Opinion poll for the support to a political party 1st poll 49% 2nd poll 51% In contrast 1st poll 75% 2nd poll 30% GED111/CDS111 Statistics in Modern Society

Quantifying Statistical Significance 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 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 Statistical significance at almost any level is no guarantee that an effect or difference is present GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Time to think Suppose an experiment finds that people taking a new herbal remedy get fewer colds than people taking a placebo, and the results are statistically significant at the 0.01 level, Has the experiment proven that the herbal remedy works? Explain. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Statistical Significance Q1-4, p236 Dose It Make Sense?? Q5-8, p236 Application Q22, 23 p237 GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Basic of Probability Outcomes The most basic possible results of observations or experiments Event A collection of one or more outcomes that share a property of interest Expressing Probability The probability of an event, expressed as P(event), is always between 0 and 1 inclusive. GED111/CDS111 Statistics in Modern Society

Theoretical Probabilities Count the total number of possible outcomes Among all possible outcomes, count the number of ways the event of interest, A, can occur. P(A) no. of ways A can occur/total no. of outcomes GED111/CDS111 Statistics in Modern Society

Relative Frequency Probabilities Repeat or observe a process many times and count the number of times the event of interest, A, occurs. P(A) No. of times A occurred/total no. of observations GED111/CDS111 Statistics in Modern Society

Subjective Probabilities Estimate probability using experience, judgment, or intuition GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Complementary Events Negation, Not A Probability Distributions Represents the probabilities of all possible events Tossing a coin 3 times, noting heads GED111/CDS111 Statistics in Modern Society

The law of Large Numbers Applies to a process for which the probability of an event A is P(A) and the results of repeated trials are independent. If the process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). The larger the number of trials, the closer the proportion should be to P(A). GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Gambler’s Fallacy The mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Does It Make Sense?? Q5-8, p256-257 GED111/CDS111 Statistics in Modern Society

Focus on Social Science Are Lotteries Fair? pp279-281 GED111/CDS111 Statistics in Modern Society