BNAD 276: Statistical Inference in Management Winter, 2015

BNAD 276: Statistical Inference in ManagementWinter, 2015 Welcome Syllabi Green sheet Seating Chart http://www.youtube.com/watch?v=Ahg6qcgoay4&watch_response

Daily group portfolios • Beginning of each lecture (first 5 minutes) • Meet in groups of 3 or 4 • Quiz one another on class material • Discuss the questions and determine the correct answer for each question • Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture • Include 4 options (a, b, c, and d) • Include a name and describe a person in a certain situation Margaret was interested in taking a Statistics course. It is likely she was interested in studying which of the following?a. economic theories of communism b. theological perspectives of life after death c. musical compositions of the 12th century d. statistical techniques and inference They can be funny or serious, and must be clear and have only one correct answer.

Please start portfolios

. • Homework AssignmentGo to D2L - Click on “Content” • Click on “Interactive Online Homework Assignments” • Complete the module: • Seven Prototypical Designs

Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Variable name is listed clearly Both axes have real numbers listed Both axes and values are labeled This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). 48 52 5660 64 68 72 1. Describe one positive correlation Draw a scatterplot (label axes) Height of Mothers (in) 2. Describe one negative correlation Draw a scatterplot (label axes) Hand in Correlation worksheet 48 52 56 60 64 68 72 76 Height of Daughters (inches) 3. Describe one zero correlation Draw a scatterplot (label axes) 4. Describe one perfect correlation (positive or negative) Draw a scatterplot (label axes) 5. Describe curvilinear relationship Draw a scatterplot (label axes)

Overview Frequency distributions The normal curve Challenge yourself as we work through characteristics of distributions to try to categorize each concept as a measure of 1) central tendency 2) dispersion or 3) shape Mean, Median, Mode, Trimmed Mean Standard deviation, Variance, Range Mean Absolute Deviation Skewed right, skewed left unimodal, bimodal, symmetric

Another example: How many kids in your family? Number of kids in family 1 4 3 2 1 8 4 2 2 14 14 4 2 1 4 2 3 2 1 8

Measures of Central Tendency(Measures of location)The mean, median and mode Mean: The balance point of a distribution. Found by adding up all observations and then dividing by the number of observations Mean for a sample: Σx / n = mean = x Mean for a population: ΣX / N = mean = µ(mu) Measures of “location” Where on the number line the scores tend to cluster Note: Σ = add up x or X = scores n or N = number of scores

Measures of Central Tendency(Measures of location)The mean, median and mode Mean: The balance point of a distribution. Found by adding up all observations and then dividing by the number of observations Mean for a sample: Σx / n = mean = x 41/ 10 = mean = 4.1 Number of kids in family 1 4 3 2 1 8 4 2 2 14 Note: Σ = add up x or X = scores n or N = number of scores

Number of kids in family 1 3 1 4 2 4 2 8 2 14 How many kids are in your family? What is the most common family size? Median: The middle value when observations are ordered from least to most (or most to least)

Number of kids in family 1 4 32 18 42 2 14 How many kids are in your family? What is the most common family size? Median: The middle value when observations are ordered from least to most (or most to least) 1, 3, 1, 4, 2, 4, 2, 8, 2, 14 1, 2, 2, 4, 2, 1, 8, 3, 4, 14

Number of kids in family 1 3 1 4 2 4 2 8 2 14 Number of kids in family 1 4 32 18 42 2 14 How many kids are in your family? What is the most common family size? Median: The middle value when observations are ordered from least to most (or most to least) 1, 3, 1, 4, 2, 4, 2, 8, 2, 14 1, 2, 2, 4, 1, 2, 2, 4, 2, 1, 2, 1, 8, 8, 3, 4, 14 3, 4, 14 2.5 2 + 3 µ=2.5 If there appears to be two medians, take the mean of the two Median always has a percentile rank of 50% regardless of shape of distribution Median also called the 2nd Quartile

Number of kids in family 1 3 1 4 2 4 2 8 2 14 Number of kids in family 1 4 32 18 42 2 14 How many kids are in your family? What is the most common family size? Median: The middle value when observations are ordered from least to most (or most to least) 1, 2, 2, 4, 2, 2, 1, 8, 3, 4, 14 3, Lower half Upper half 2.5 1, 2, 4, 2, 1, 8, 3, 3, 14 2nd Quartile Middle number of all scores (Median) 4, 4, 3, 8, 2, 2, 4, 4, 14 1, 2, 1, 2, 3rd Quartile Middle number of upper half of scores 1st Quartile Middle number of lower half of scores

Mode: The value of the most frequent observation Score f . 1 2 2 3 3 1 4 2 5 0 6 0 7 0 8 1 9 0 10 0 11 0 12 0 13 0 14 1 Number of kids in family 1 3 1 4 2 4 2 8 2 14 Please note: The mode is “2” because it is the most frequently occurring score. It occurs “3” times. “3” is not the mode, it is just the frequency for the value that is the mode Bimodal distribution: If there are two most frequent observations

What about central tendency for qualitative data? Mode is good for nominal or ordinal data Median can be used with ordinal data Mean can be used with interval or ratio data

Overview Frequency distributions The normal curve Challenge yourself as we work through characteristics of distributions to try to categorize each concept as a measure of 1) central tendency 2) dispersion or 3) shape Mean, Median, Mode, Trimmed Mean Skewed right, skewed left unimodal, bimodal, symmetric

A little more about frequency distributions An example of a normal distribution

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Normal distribution In all distributions: mode = tallest point median = middle score mean = balance point In a normal distribution: mode = mean = median

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Positively skewed distribution In all distributions: mode = tallest point median = middle score mean = balance point In a positively skewed distribution: mode < median < mean Note: mean is most affected by outliers or skewed distributions With Bill Gates our Average Income would be $38 million a year

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Negatively skewed distribution In all distributions: mode = tallest point median = middle score mean = balance point In a negatively skewed distribution: mean < median < mode Note: mean is most affected by outliers or skewed distributions

Mode: The value of the most frequent observation Bimodal distribution: Distribution with two most frequent observations (2 peaks) Example: Ian coaches two boys baseball teams. One team is made up of 10-year-olds and the other is made up of 16-year-olds. When he measured the height of all of his players he found a bimodal distribution

Overview Frequency distributions The normal curve Mean, Median, Mode, Trimmed Mean Standard deviation, Variance, Range Mean Absolute Deviation Skewed right, skewed left unimodal, bimodal, symmetric

Frequency distributions The normal curve

Variability What might this be? Some distributions are more variable than others Let’s say this is our distribution of heights of men on U of A baseball team 5’ 7’ 6’ 6’6” 5’6” 5’ 7’ 6’ 6’6” 5’6” Mean is 6 feet tall What might this be? 5’ 7’ 6’ 6’6” 5’6”

Dispersion: Variability 5’ 7’ 6’ 6’6” 5’6” 5’ 7’ 6’ 6’6” 5’6” 5’ 7’ 6’ 6’6” 5’6” Some distributions are more variable than others The larger the variability the wider the curve tends to be The smaller the variability the narrower the curvetends to be A B Range: The difference between the largest and smallest observations C Range for distribution A? Range for distribution B? Range for distribution C?

Wildcats Basketball team: Tallest player = 84” (same as 7’0”)(Kaleb Tarczewskiand DusanRistic) Shortest player = 70” (same as 5’10”) (Parker Jackson-Cartwritght) Fun fact: Mean is 78 Range: The difference between the largest and smallest scores 84” – 70” = 14” xmax - xmin = Range Range is 14”

Baseball Fun fact: Mean is 72 Wildcats Baseball team: Range: The difference between the largest and smallest score 77” – 69” = 8” Tallest player = 77” (same as 6’5”) (Austin Schnabel) Shortest player = 69” (same as 5’9”) (Justin Behnke and Ernie DeLaTrinidad ) xmax - xmin = Range Range is 8”(77” – 69” ) Please note: No reference is made to numbers between the min and max

Variability Standard deviation: The average amount by which observations deviate on either side of their mean Generally, (on average) how far away is each score from the mean? Mean is 6’

Let’s build it up again…U of A Baseball team Deviation scores Diallo is 0” Diallo is 6’0” Diallo’s deviation score is 0 6’0” – 6’0” = 0 Diallo 5’8” 5’10” 6’0” 6’2” 6’4”

Deviation scores Diallo is 0” Let’s build it up again…U of A Baseball team Preston is 2” Diallo is 6’0” Diallo’s deviation score is 0 Preston is 6’2” Preston Preston’s deviation score is 2” 6’2” – 6’0” = 2 5’8” 5’10” 6’0” 6’2” 6’4”

Deviation scores Diallo is 0” Let’s build it up again…U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Diallo is 6’0” Diallo’s deviation score is 0 Hunter Preston is 6’2” Preston’s deviation score is 2” Mike Mike is 5’8” Mike’s deviation score is -4” 5’8” – 6’0” = -4 5’8” 5’10” 6’0” 6’2” 6’4” Hunter is 5’10” Hunter’s deviation score is -2” 5’10” – 6’0” = -2

Deviation scores Diallo is 0” Let’s build it up again…U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Diallo’s deviation score is 0 David Preston’s deviation score is 2” Mike’s deviation score is -4” Shea Hunter’s deviation score is -2” Shea is 6’4” Shea’s deviation score is 4” 5’8” 5’10” 6’0” 6’2” 6’4” 6’4” – 6’0” = 4 David is 6’ 0” David’s deviation score is 0 6’ 0” – 6’0” = 0

Deviation scores Diallo is 0” Let’s build it up again…U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Diallo’s deviation score is 0 David Preston’s deviation score is 2” Mike’s deviation score is -4” Shea Hunter’s deviation score is -2” Shea’s deviation score is 4” David’s deviation score is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Deviation scores Diallo is 0” Let’s build it up again…U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Deviation scores Deviation scores: The amount by which observations deviate on either side of their mean (x- µ) (x- µ) Diallo is 0” Preston is 2” How far away is each score from the mean? Mike is -4” Hunter is -2 Shea is 4 Mean David is 0” Diallo Deviation score Mike Preston (x - µ) = ? Shea Hunter Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Diallo How do we find each deviation score? (x- µ) Preston Hunter Diallo Mike Preston Find distance of each person from the mean (subtract their score from mean)

Deviation scores Deviation scores: The amount by which observations deviate on either side of their mean (x- µ) (x- µ) Diallo is 0” Preston is 2” How far away is each score from the mean? Mike is -4” Hunter is -2 Shea is 4 Mean David is 0” Diallo Deviation score Preston (x - µ) = ? Shea Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Remember It’s relative to the mean Based on difference from the mean

Σ(x - x) = 0 Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean (x- µ) Diallo is 0” Preston is 2” Mike is -4” How far away is each score from the mean? Hunter is -2 Shea is 4 Mean David is 0” Add up Deviation scores Diallo Preston Σ(x - µ) = ? Shea Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” How do we find the average height? = average height Σx N How do we find the average spread? Σ(x - µ) = average deviation Σ(x - µ) = 0 N

Σ(x - x) = 0 Σ(x - x) Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean (x- µ) Diallo is 0” Preston is 2” Mike is -4” How far away is each score from the mean? Hunter is -2 Shea is 4 Mean David is 0” Diallo Preston Σ(x - µ) = ? Shea 2 Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Square the deviations Big problem 2 Σ(x - µ) 2 Σ(x - µ) Σ(x - µ) = 0 N

Standard deviation: The average amount by which observations deviate on either side of their mean These would be helpful to know by heart – please memorize these formula

Standard deviation: The average amount by which observations deviate on either side of their mean What do these two formula have in common?

Standard deviation: The average amount by which observations deviate on either side of their mean “n-1” is Degrees of Freedom” How do these formula differ?

BNAD 276: Statistical Inference in Management Winter, 2015