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Binomial Distribution Fundamentals in Statistics

Learn about the binomial distribution, its mean, variance, and examples. Explore hypothesis testing and significance levels in statistics lectures. Real-life examples like childbirth and genetics are covered.

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Binomial Distribution Fundamentals in Statistics

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  1. The Binomial Distribution Nina Gunnes October 17, 2019 Lecture 5

  2. Count variable • Resulting from counting the number of times an event occurs • Variable only taking on the value zero and positive integer values • Mean (expected value) • Center of the probability distribution • Variance and standard deviation • Measure of the dispersion, or spread, of the distribution • and Lecture 5

  3. Example • Norwegian National Advisory Unit on Women’s Health • 25 persons, of which eight are pet owners (and 17 are not) • : number of pet owners • Distribution of pet owners in a random sample of three persons Lecture 5

  4. Binomial experiment • Consisting of a sequence of repeated trials • Each trial resulting in one of two possible outcomes • (for success) • (for failure) • Probability of success (failure) being the same in each trial • Denoted by () • Trials being independent of each other • Outcome of one trial not affecting outcome of other trials Lecture 5

  5. Examples (Aalen et al., 2006) • Childbirth • Boy or girl • Throw of the die • A six or not • Treatment with penicillin • Anaphylactic shock or not • Gallup poll • Voting for a political party or not • Occurrence of disease • Suffering from allergies or not • Genetics • Sickle cell anemia or not Lecture 5

  6. Binomial distribution • Considering a binomial experiment of trials • Two possible outcomes: (success) or (failure) • : number of times occurres (i.e., number of successes) • : probability of success in each trail • A hypothetical case of successes: • Probability of this sequence: • Number of ways to arrange the order of this sequence: , where • Probability of successes: Lecture 5

  7. Binomial distribution, cont. • Mean (expected value) • Expected number of successes • Variance • Skewed distribution when is close to 0 or 1 • More symmetrical distribution for central values of • Perfect symmetry for (i.e., success and failure equally likely) Lecture 5

  8. Binomial distribution, cont. Lecture 5

  9. Example • Flipping a coin 10 times () • Two possible outcomes of each trial: head and tail • Probability of head in each trial: • : number of heads • Probability of at least nine heads? Lecture 5

  10. Hypotheses blah, blah, blah …, significance level blah, blah, blah … Hypothesis testing • Drawing conclusions from data subject to random variation • Evaluating an intervention, comparing treatments, etc. • Is the observed result indicating an effect (or difference)? • Is the observed result merely an act of chance? • Defining a null hypothesis • : no effect (or no difference) • Defining an alternative hypothesis • : effect (or difference) – possibly in a specific direction Yeah, yeah, but is the new treatment better than the old one or not?! Lecture 5

  11. Hypothesis testing, cont. • Different alternative hypotheses for one-sided and two-sided tests • One-sided : one treatment more (less) efficacious than the other • Two-sided : one treatment either more or less efficacious than the other • Direction of any potential difference usually not known in advance • Two-sided test most common in practice • Two types of possible errors • Type I error: erroneously rejecting when it is in fact true • Type II error: erroneously accepting when it is in fact false Lecture 5

  12. Hypothesis testing, cont. • Significance level expressing necessary certainty of the conclusion • Denoted by • Equal to the probability of making a type I error • Serious consequences of rejecting : choosing a low value of (e.g., 1%) • No serious consequences of rejecting : choosing a high value of (e.g., 5%) • Calculating the significance probability, or p value, of a test statistic • Probability of values at least as extreme as the observed value under • (whether one-sided or two-sided) determining what is deemed extreme Lecture 5

  13. Hypothesis testing, cont. • Comparing the p value to the significance level • Small value () • Observed result not consistent with • Rejecting in favor of (that is, accepting ) • Large value () • Failing to reject – NOT(!) the same as proving that is true • Ideally, choosing , , and before the study outcome is known Lecture 5

  14. Migraine (Aalen et al., 2006) • Testing a new medication to treat migraine • : traditional medication • : new medication • Randomized double-blind crossover study • One month of treatment with and one month of treatment with • Neither patient nor physician knowing which medication is taken when • When is the patient least affected by migraine? • Month corresponding to treatment with or ? https://pxhere.com/en/photo/1448737 Lecture 5

  15. Migraine (Aalen et al., 2006), cont. • Eight patients included in a hypothetical study • : number of patients preferring over • binomially distributed with trials and unknown probability • Seven out of the eight patients most comfortable with • Indication of being the best medication or just a coincidence? • Defining the hypotheses for a one-sided test • : (the two medications equally efficacious) • : ( more efficacious than ) Lecture 5

  16. Migraine (Aalen et al., 2006), cont. • Significance probability for a one-sided test • Rejecting at a significance level of 5% • more efficacious than • Defining the alternative hypothesis for a two-sidedtest • : (either more or less efficacious than ) Lecture 5

  17. Migraine (Aalen et al., 2006), cont. • Significance probability for a two-sided test • Not rejecting at a significance level of 5% • No basis for claiming that is more efficacious than Lecture 5

  18. Cot death (Aalen et al., 2006) • Also known as sudden infant death syndrome (SIDS) • SIDS epidemic in Norway and other Western countries in the 1980s • Related to the child’s sleeping position? • 11 children died of SIDS in Hordaland county in 1990 • 10 of the children were in a front-sleeping position (“mageleie”) • Front sleeping in 15% of children in Hordaland county on the average • : number of SIDS babies in a front-sleeping position • binomially distributed with trials and unknown probability Lecture 5

  19. Cot death (Aalen et al., 2006), cont. • Is the observed result conspicuous? • Hypotheses for a one-sided test • : (same front-sleeping proportion as the general population) • : (greater front-sleeping proportion than the general population) • Significance probability for a one-sided test • () Lecture 5

  20. Cot death (Aalen et al., 2006), cont. • Rejecting at a significance level of 5% • Strikingly high frequency of front sleeping among the 11 SIDS babies • Front-sleeping position no longer recommended for babies • Placing babies on their backs to sleep instead https://pixabay.com/vectors/baby-girl-cartoons-pink-new-baby-33252/ Lecture 5

  21. References • Aalen OO, Frigessi A, Moger TA, Scheel I, Skovlund E, Veierød MB. 2006. Statistiskemetoderimedisinoghelsefag. Oslo: Gyldendal akademisk. Lecture 5

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