Lecture #14 Thursday, October 6, 2016 Textbook: Sections 8.4, 8.5, 8.6 Statistics 200 Lecture #14 Thursday, October 6, 2016 Textbook: Sections 8.4, 8.5, 8.6 Objectives: • Recognize the four conditions for a binomial random variable • Calculate the mean and standard deviation for a binomial random variable • Use probability notation for continuous random variables and relate this notation to area under a density function. • Standardize any normal distribution and then use tables or a computer to find probabilities

Example: What do you notice? pick a card from shuffled deck Look for an ace Put card back in the deck; reshuffle the deck repeat three more times X: number of aces in 4 tries

A binomial random variable is: X = number of ___________ in n ________________ trials of a random circumstance in which p = probability of success is __________. successes independent constant In other words, X counts the successes in a __________________. binomial experiment Your job: Recognize when an experiment satisfies the 4 conditions for a binomial experiment. These 4 conditions are…

Conditions for a binomial experiment 1 There are n “trials”, where n is fixed and known in advance 2 We can define two possible outcomes for each trial: “Success” (S) and “Failure” (F) 3 The outcomes are independent; no single outcome influences any other outcome 4 The probability of “Success” is the same for each trial. We use “p” to write P(Success).

If X= # of aces in 4 tries, is X a binomial random variable? Must confirm that our set-up satisfies all four conditions. Yes: n=4 Fixed # trials: __________ 1 S / F outcomes: ____________ Yes: S = ace Conclusion : 2 Yes Independent trials: _____ 3 Yes! n = 4, p = 4/52 Yes P(success) constant: _____ 4

Slight Change: Is this still a Binomial? pick a card from shuffled deck Look for an ace repeat three more times Put card aside; reshuffle the deck X: number of aces in 4 tries

If X= # of aces in 4 tries, is X a binomial random variable? Must confirm that our set-up satisfies all four conditions. Yes: n=4 Fixed # trials: __________ 1 S / F outcomes: ____________ Yes: S = ace Conclusion : 2 No Independent trials: _____ 3 NOT a binomial No P(success) constant: _____ 4

Review of confidence intervals Suppose we take a sample of 1009 adults and ask them a yes or no question. Of them, 646 answer yes, and the rest answer no. What is the value of p-hat? 646 – 363 646 / 363 1009 – 646 646 / 1009 1/sqrt(1009)

Review of confidence intervals 2 Suppose we take a sample of 1009 adults and ask them a yes or no question. Of them, 646 answer yes, and the rest answer no. What is the value of the margin of error for a 95% confidence interval? 646 – 363 646 / 363 1009 – 646 646 / 1009 1/sqrt(1009)

Americans' Coffee Consumption Is Steady, Few Want to Cut Back Example: Americans' Coffee Consumption Is Steady, Few Want to Cut Back • Initial Survey Question: How many cups of coffee, if any, do you drink on an average day? Coffee shops are reportedly the fastest-growing segment of the restaurant industry, yet the percentage of Americans who regularly drink coffee hasn't budged. Sixty-four percent of U.S. adults report drinking at least one cup of coffee on an average day, unchanged from 2012. Results for this Gallup poll are based on telephone interviews conducted July 8-12, 2015, with a random sample of 1,009 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia.

Is the Gallup polling process a binomial experiment? check! 1. n = 1009 trials 2. Success = “drink at least one cup of coffee a day” Failure = “don’t drink any coffee” 3. Independent trials: 4. p remains constant: Conclusion: (is a binomial) (not a binomial) Thus, if X = number who drink at least one cup of coffe a day in a sample of 1009 U.S. national adults, X is a binomial random variable. check! close enough! (We don’t sample with replacement but the population is huge)

Which binomial condition is not met? • An airplane flight is considered on time if it arrives within 15 minutes of its scheduled arrival time. At O’Hara Airport, in Chicago, 300 flights are scheduled to arrive on one day in January. X = number of the 300 flights that arrive on time. n = ________ success = ______ independent trials: _____ same probability for each trial: _____________ 300 on time arrival no! probably not

Which binomial condition is not met? • A football team plays 12 games in its regular season where it is determined whether or not the team wins each game. X = number of games the team wins during the regular season of 12 games n = ________ success = ______ independent trials: _____ same probability for each trial: _____________ 12 winning game doubtful no!

Which binomial condition is not met? • A woman buys a lottery ticket every week. She continues to buy tickets until she wins. Let X = number of tickets that she buys until she finally wins. n = ________ success = ______________ independent trials: _____ same probability for each trial: _____________ ??? winning lottery yes yes

Mean and standard deviation for binomial random variables

Final example Consider our 3-coin example with X = # heads. This is a binomial random variable with n = ____, p = ______ Thus the mean number of heads is _______ = ________ The standard deviation is _______________ = _______________ 3 0.5 3 × 0.5 1.5 0.866

Continuous Random Variable Assumes a range of values covering an interval. _____________. May be limited by instrument’s accuracy / decimal points, but still continuous. Find probabilities using a probability density function, which is a curve. Calculate probabilities by finding the area under the curve. is this area • We can’t find probabilities for exact outcomes. • For example: P(X = 2) = 0. • Instead we can find probabilities for a range of values.

Probability density function Recall – we calculate probabilities by finding area under the curve. This is the density for a chi-square random variable. The density is larger for smaller values of X. To calculate a probability, we must find an __________. area

Probability density function Recall – we calculate probabilities by finding area under the curve. Red area here is P(X>1.5) The shaded part has area _________ The area under the entire curve is __________ 0.2207 1.0

= 0.15 = 0.68 = 0.05 = 0.79

An important probability P(X=5) = ________ A ______ has ___ area line no Rule: P( X=a ) = ___ for any value a

Return to the normal distribution We’ve already seen the normal distribution Mean Standard deviation Empirical rule Used Empirical Rule to make histogram

Goal: Standardization Limitless number of Normal Distributions One Standard Normal Distribution

Normal Distributions: Bell-Shape (General) Normal: General normal distribution: Standard normal distribution:

How to find Normal Probabilities Use calculus – integrate Read normal probability tables Use a probability calculator Minitab internet http://davidmlane.com/normal.html 1 Total area under the curve = _____

http://davidmlane.com/normal.html Default Screen Probability Density Function Forward Backwards

What is the probability that Z is: greater than 1? at least 1? exactly 1?

0.16 P(Z > 1) = Forward

Minitab: Graph> Probability Distribution Plots

Minitab: (Density) Probability Distribution Plot P(Z > 1) = 0.1587

0.16 0.16 What is the probability that Z is: greater than 1? at least 1? exactly 1? 0.16 0.16

Standard normal example: Z is a standard normal random variable. Which is the entirely correct picture for: P(Z > 1.72)?

Clicker Question of understanding: Z is a standard normal random variable. Consider the probability statement P(-1.5 < Z < 2.0). Which is the only possible probability for this statement? 0.6800 B. 0.9104 C. 0.9500 D. 0.9654 E 0.9970

How to relate all this to Z-scores We can standardize values from any normal distribution to relation them to the standard normal distribution.

-1 Z-score 1: P(7<X<10) = P( __ < Z < ___ ) Z-score 2: Thus, P(7<X<10) = P( __ < Z < ___ ) -1 Z-score 2:

If you understand today’s lecture… 8.11, 8.41, 8.43, 8.47, 8.49, 8.63, 8.65, 8.67, 8.71, 8.81a, 8.83, 8.85 (for the normal distribution problems, sketch a picture!) Objectives: • Recognize the four conditions for a binomial random variable • Calculate the mean and standard deviation for a binomial random variable • Use probability notation for continuous random variables and relate this notation to area under a density function. • Standardize any normal distribution and then use tables or a computer to find probabilities