Social Science Reasoning Using Statistics Psychology 138 2019

Announcements Exam 2 is Wed. March 6th Quiz 4 Fri March 1nd Covers z-scores, Normal distribution, and describing correlations Announcements

Outline Transformations: z-scores Normal Distribution Using Unit Normal Table Today’s lecture puts lots of stuff together: Probability Frequency distribution tables Histograms Z-scores Today Start the quincnux machine Outline

Flipping a coin example Number of heads HHH 3 HHT 2 HTH 2 HTT 1 THH 2 THT 1 TTH 1 TTT Flipping a coin example

Flipping a coin example Number of heads 3 2 Number of heads 1 2 3 .1 .2 .3 .4 probability X f p 3 1 .125 2 .375 2 .375 .375 1 .125 .125 2 1 1 Flipping a coin example

Flipping a coin example What’s the probability of flipping three heads in a row? .4 .3 probability .2 p = 0.125 .1 .125 .375 .375 .125 Think about the area under the curve as reflecting the proportion/probability of a particular outcome 1 2 3 Number of heads Flipping a coin example

Flipping a coin example What’s the probability of flipping at least two heads in three tosses? .4 .3 probability .2 p = 0.375 + 0.125 = 0.50 .1 .125 .375 .375 .125 1 2 3 Number of heads Flipping a coin example

Flipping a coin example What’s the probability of flipping all heads or all tails in three tosses? .4 .3 probability .2 p = 0.125 + 0.125 = 0.25 .1 .125 .375 .375 .125 1 2 3 Number of heads Flipping a coin example

Flipping a coin example Coin flipping results in The Binomial Distribution As you flip more coins, N = 20 Flipping a coin example

Flipping a coin example Coin flipping results in The Binomial Distribution As you flip more coins, the binomial distribution can be approximated by the Normal Distribution N = 65 Flipping a coin example

The Normal Distribution (Sometimes called the “Bell Curve”) Many commonly occurring distributions in nature are approximately Normal Originally called the curve of normal errors Normal Distribution

The Normal Distribution (Sometimes called the “Bell Curve”) Many commonly occurring distributions in nature are approximately Normal Common approximate empirical distribution for deviations around a continually scaled variable (errors of measurement) Defined by density function (area under curve) for variable X given μ & σ2 Symmetrical & unimodal; Mean = median = mode Converting to standardized z-scores Mean = 0 ±1 σ are inflection points of curve (change of direction) Originally called the curve of normal errors Normal Distribution

The Normal Distribution (Sometimes called the “Bell Curve”) Many commonly occurring distributions in nature are approximately Normal Common approximate empirical distribution for deviations around a continually scaled variable (errors of measurement) Use calculus to find areas under curve (rather than frequency of a score) We will use a table rather to find the probabilities rather than do the calculus. Galileo Originally called the curve of normal errors Gauss Check out the quincnux machine Normal Distribution

The Normal Distribution (Sometimes called the “Bell Curve”) Important landmarks in the distribution (and the areas under the curve) %(μ to 1σ) = 34.13 p(μ < X < 1σ) + p(μ > X > -1σ) ≈ .68 %(1σ to 2σ) = 13.59 p(1σ < X < 2σ) + p(-1σ > X > -2σ) = 27%, cumulative = .95 %(2σ to ∞) = 2.28 p(X > 2σ) + p(X < -2σ) ≈ 5%, cumulative = 1.00 We will use the unit normal table rather to find other probabilities 68% 34.13 2.28 Originally called the curve of normal errors 95% 13.59 100% Normal Distribution

Lots of places to get the Unit Normal Table information But be aware that there are many ways to organize the table, it is important to understand the table that you use Unit normal table in your reading packet And online: http://psychology.illinoisstate.edu/jccutti/psych138/resources copy/TABLES.HTMl - ztable “Area Under Normal Curve” Excel tool (created by Dr. Joel Schneider) Bell Curve iPhone app Do a search on “Normal Table” in Google. Resources and tools

Unit Normal Table (linked online in labs) X f p cp 3 1 .125 1.0 2 .375 .875 .500 |z| .00 .01 : 1.0 2.0 3.0 0.5000 0.1587 0.0228 0.0013 0.4960 0.1562 0.0222 Number of heads 1 2 3 .1 .2 .3 .4 probability .125 .875 We will use the unit normal table rather to find other probabilities Unit Normal Table (linked online in labs)

Unit Normal Table (in textbook) z p Proportions beyond z-scores Same p-values for + and - z-scores p-values = 0.50 to 0.0013 |z| In body In tail 0.00 : 1.00 2.0 3.0 0.5000 0.8413 0.9772 0.9987 0.1587 0.0228 0.0013 tail body “body” is defined as the larger part of the distribution Note. : indicates skipped rows Unit Normal Table (in textbook)

Unit Normal Table (in textbook) z p Proportions beyond z-scores Same p-values for + and - z-scores p-values = 0.50 to 0.0013 |z| In body In tail 0.00 : 1.00 2.0 3.0 0.5000 0.8413 0.9772 0.9987 0.1587 0.0228 0.0013 .1359+.0228 = .1587 For z = 1.00 , 15.87% beyond, p(z > 1.00) = 0.1587 1.01, 15.62% beyond, p(z > 1.00) = 0.1562 Note. : indicates skipped rows Unit Normal Table (in textbook)

Unit Normal Table (in textbook) Proportions beyond z-scores Same p-values for + and - z-scores p-values = 0.50 to 0.0013 |z| In body In tail 0.00 : 1.00 2.0 3.0 0.5000 0.8413 0.9772 0.9987 0.1587 0.0228 0.0013 For z = -1.00 , 15.87% beyond, p(z < -1.00) = 0.1587 100% - 15.87% = 84.13% = 0.8413 Note. : indicates skipped rows Unit Normal Table (in textbook)

Unit Normal Table (in textbook) Proportions beyond z-scores Same p-values for + and - z-scores p-values = 0.50 to 0.0013 |z| In body In tail 0.00 : 1.00 2.0 3.0 0.5000 0.8413 0.9772 0.9987 0.1587 0.0228 0.0013 For z = 2.00, 2.28% beyond, p(z > 2.00) = 0.0228 2.01, 2.22% beyond, p(z > 2.01) = 0.0222 As z increases, p decreases Note. : indicates skipped rows Unit Normal Table (in textbook)

Unit Normal Table (in textbook) Proportions beyond z-scores Same p-values for + and - z-scores p-values = 0.50 to 0.0013 |z| In body In tail 0.00 : 1.00 2.0 3.0 0.5000 0.8413 0.9772 0.9987 0.1587 0.0228 0.0013 For z = -2.00, 2.28% beyond, p(z < -2.00) = 0.0228 Note. : indicates skipped rows Unit Normal Table (in textbook)

Unit Normal Table cumulative version (in some other books) z .00 .01 -3.4 : -1.0 1.0 3.4 0.0003 0.1587 0.5000 0.8413 0.9997 0.1562 0.5040 0.8438 Proportions left of z-scores: cumulative Requires table twice as long p-values 0.0003 to 0.9997 50% + 34.13 = 84.13% to the left Cumulative % of population starting with the lowest value For z = +1, Unit Normal Table cumulative version (in some other books)

SAT examples Population parameters of SAT: μ = 500, σ = 100, normally distributed Example 1 Suppose you got 630 on SAT. What % who take SAT get your score or better? From table: z(1.3) =.0968 9.68% above this score Solve for z-value of 630. Find proportion of normal distribution above that value. Hint: I strongly suggest that you sketch the problem to check your answer against SAT examples

SAT examples Population parameters of SAT: μ = 500, σ = 100, normally distributed Example 2 Suppose you got 630 on SAT. What % who take SAT get your score or worse? From table: z(1.3) =.0968 100% - 9.68% = 90.32% below score (percentile) Solve for z-value of 630. Find proportion of normal distribution below that value. SAT examples

Wrap up In lab Questions? Using the normal distribution Check the quincnux machine Brandon Foltz: Understanding z-scores (~22 mins) StatisticsFun (~4 mins) Chris Thomas. How to use a z-table (~7 mins) Dr. Grande. Z-scores in SPSS (~7 mins) Wrap up