Top Ten #1

Descriptive Statistics

NOTE! This Power Point file is not an introduction, but rather a checklist

of topics to review

Location: central tendency

• Population Mean =µ= Σx/N = (5+1+6)/3 = 12/3 = 4

• Algebra: Σx = N*µ = 3*4 =12

• Do NOT use if N is small and extreme values

• Ex: Do NOT use if 3 houses sold this week, and one was a mansion

Location

• Median = middle value

• Ex: 5,1,6

• Step 1: Sort data: 1,5,6

• Step 2: Middle value = 5

• OK even if extreme values

• Home sales: 100K,200K,900K, so

mean =400K, but median = 200K

Location

• Mode: most frequent value

• Ex: female, male, female

• Mode = female

• Ex: 1,1,2,3,5,8: mode = 1

Relationship

• Case 1: if symmetric (ex bell, normal), then mean = median = mode

• Case 2: if positively skewed to right, then mode<median<mean

• Case 3: if negatively skewed to left, then mean<median<mode

Dispersion

• How much spread of data

• How much uncertainty

• Range = Max-Min > 0

• But range affected by unusual values

• Ex: Santa Monica = 105 degrees once a century, but range would be 105-min

Standard Deviation

• Better than range because all data used

• Population SD = Square root of variance =sigma =σ

• SD > 0

Empirical Rule

• Applies to mound or bell-shaped curves

• Ex: normal distribution

• 68% of data within + one SD of mean

• 95% of data within + two SD of mean

• 99.7% of data within + three SD of mean

Sample Variance

2

1)(

nxx

Standard deviation = Square root of variance

1)( 2

nxx

s

Sample Standard Deviation

X

6 6-8=-2 (-2)(-2)= 4

6 6-8=-2 4

7 7-8=-1 (-1)(-1)= 1

8 8-8=0 0

13 13-8=5 (5)(5)= 25

Sum=40 Sum=0 Sum = 34

Mean=40/5=8

xx 2)( xx

Standard Deviation

Total variation = 34

• Sample variance = 34/4 = 8.5

• Sample standard deviation =

square root of 8.5 = 2.9

Graphical Tools

• Line chart: trend over time• Scatter diagram: relationship between two

variables• Bar Chart: frequency for each category• Histogram: frequency for each class of

measured data (graph of frequency distr)• Box Plot: graphical display based on

quartiles, which divide data into 4 parts

Top Ten #2

• Hypothesis Testing

Ho: Null Hypothesis

• Population mean=µ

• Population proportion=π

• Never include sample statistic in hypothesis

HA: Alternative Hypothesis

• ONE TAIL ALTERNATIVE– Right tail: µ>number(smog ck)

π>fraction(%defectives)

Left tail: µ<number(weight in box of crackers)

π<fraction(unpopular President’s % approval low)

Two-tail Alternative

• Population mean not equal to number (too hot or too cold)

• Population proportion not equal to fraction(% alcohol too weak or too strong)

Reject null hypothesis if

• Absolute value of test statistic > critical value• Reject Ho if |Z Value| > critical Z• Reject Ho if | t Value| > critical t• Reject Ho if p-value < significance level (note that

direction of inequality is reversed)• Reject Ho if very large difference between sample

statistic and population parameter in Ho

Example: Smog Check

• Ho: µ = 80• HA: µ > 80• If test statistic =2.2 and critical value =

1.96, reject Ho, and conclude that the population mean is likely > 80

• If test statistic = 1.6 and critical value = 1.96, do not reject Ho, and reserve judgment about Ho

Type I vs Type II error

• Alpha=α = P(type I error) = Significance level = probability you reject true null hypothesis

• Ex: Ho: Defendant innocent• α = P(jury convicts innocent person)• Beta= β = P(type II error) = probability you do not

reject a null hypothesis, given Ho false

• β =P(jury acquits guilty person)

Type I vs Type II Error

Ho true Ho false

Reject Ho Alpha =α =

P(type I error)

1 – β

Do not reject Ho 1-α Beta =β =

P(type II error)

Top Ten #3

• Confidence Intervals:

Mean and Proportion

Confidence Interval: Mean

• Use normal distribution (Z table if):

population standard deviation (sigma) known and either (1) or (2):

(1) Normal population

(2) Sample size > 30

Confidence Interval: Mean

• If normal table, then µ =(Σx/n)+ Z(σ/n1/2), where n1/2 is the square root of n

Normal table

• Tail = .5(1 – confidence level)

• NOTE! Different statistics texts have different normal tables

• This review uses the tail of the bell curve

• Ex: 95% confidence: tail = .5(1-.95)= .025

• Z.025 = 1.96

Example

• n=49, Σx=490, σ=2, 95% confidence

• µ = (490/49) + 1.96(2/7) = 10 + .56

• 9.44 < µ < 10.56

Conf. Interval: Mean t distribution

• Use if normal population but population standard deviation (σ) not known

• If you are given the sample standard deviation (s), use t table, assuming normal population

• If one population, n-1 degrees of freedom

t distribution

• µ = (Σx/n) + tn-1(s/n1/2)

Conf. Interval: Proportion

• Use if success or failure

(ex: defective or ok)

Normal approximation to binomial ok if

(n)(π) > 5 and (n)(1-π) > 5, where

n = sample size

π= population proportion

NOTE! NEVER use the t table if proportion!!

Confidence Interval: proportion

• Π= p + Z(p(1-p)/n)1/2

• Ex: 8 defectives out of 100, so p = .08 and

n = 100, 95% confidence

Π= .08 + 1.96(.08*.92/100)1/2

= .08 + .05

Interpretation

• If 95% confidence, then 95% of all confidence intervals will include the true population parameter

• NOTE! Never use the term “probability” when estimating a parameter!! (ex: Do NOT say ”Probability that population mean is between 23 and 32 is .95” because parameter is not a random variable)

Point vs Interval Estimate

• Point estimate: statistic (single number)

• Ex: sample mean, sample proportion

• Each sample gives different point estimate

• Interval estimate: range of values

• Ex: Population mean = sample mean + error

• Parameter = statistic + error

Width of Interval

• Ex: sample mean =23, error = 3

• Point estimate = 23

• Interval estimate = 23 + 3, or (20,26)

• Width of interval = 26-20 = 6

• Wide interval: Point estimate unreliable

Wide interval if

• (1) small sample size(n)• (2) large standard deviation(σ)• (3) high confidence interval (ex: 99% confidence

interval wider than 95% confidence interval)

If you want narrow interval, you need a large sample size or small standard deviation or low confidence level.

Top Ten #4: Linear Regression

• Regression equation: y=bo+b1x

• y=dependent variable=predicted value

• x= independent variable

• bo=y-intercept =predicted value of y if x=0

• b1=slope=regression coefficient

=change in y per unit change in x

Slope vs correlation

• Positive slope (b1>0): positive correlation between x and y (y incr if x incr)

• Negative slope (b1<0): negative correlation (y decr if x incr)

• Zero slope (b1=0): no correlation(predicted value for y is mean of y), no linear relationship between x and y

Simple linear regression

• Simple: one independent variable, one dependent variable

• Linear: graph of regression equation is straight line

Coefficient of determination

• R2 = % of total variation in y that can be explained by variation in x

• Measure of how close the linear regression line fits the points in a scatter diagram

• R2 = 1: max possible value: perfect linear relationship between y and x (straight line)

• R2 = 0: min value: no linear relationship

example

• Y = salary (female manager, in thousands of dollars)

• X = number of children

• n = number of observations

Given data

x y

2 48

1 52

4 33

Totals

x y

2 48

1 52

4 33 n=3

Sum=7 Sum=133

3.237 x

3.443

133 y

Slope = -6.500

• Method of Least Squares formulas not on 301 exam

• B1 = -6.500 given

Interpret slope

If one female manager has 1 more child than another, salary is $6500

lower

Intercept

bo= y – b1x

Intercept

bo=44.33-(-6.5)(2.33) = 59.5

Interpret intercept

If number of children is zero, expected salary is $59,500

Regression Equation

• Y = 59.5 – 6.5X

Forecast salary if 3 children

59.5 –6.5(3) = 40

$40,000 = expected salary

averagey

actualy

xbboforecasty 1ˆ

yyerror ˆ

2)ˆ( 2

nyy

S

Standard error

(1)=x (2)=y (3)

59.5-6.5x

(4)=

(2)-(3)

2 48 46.5 1.5 2.25

1 52 53 -1 1

4 33 33.5 -.5 .25

SSE=3.5

y 2)ˆ( yy

9.15.323

5.3

S

Interpret

Actual salary typically $1900 away from expected salary

• Sources of Variation (V)

• Total V = Explained V + Unexplained V

• SS = Sum of Squares = V

• Total SS = Regression SS + Error SS

• SST = SSR + SSE

• SSR = Explained V, SSE = Unexplained

Coefficient of Determination

• R2 = SSR SST

• R2 = 197 = .98 200.5

• Interpret: 98% of total variation in salary can be explained by variation in number of children

0 < R2 < 1

• 0: No linear relationship since SSR=0 (explained variation =0)

• 1: Perfect relationship since SSR = SST (unexplained variation = SSE = 0), but does not prove cause and effect

R=Correlation Coefficient

• Case 1: slope < 0

• R < 0

• R is negative square root of coefficient of determination

2RR

Our Example

• Slope = b1 = -6.5

• R2 = .98

• R = -.99

Case 2: Slope > 0

• R is positive square root of coefficient of determination

• Ex: R2 = .49

• R = .70

• R has no interpretation

• R overstates relationship

Caution

• Nonlinear relationship (parabola, hyperbola, etc) can NOT be measured by R2

• In fact, you could get R2=0 with a nonlinear graph on a scatter diagram

R=correlation coefficient

• Case 1: If b1>0, R is the positive square root of the coefficient of determination

• Ex#1: y = 4+3x, R2=.36: R = +.60• Case 2: If b1<0, R is the negative square

root of the coefficient of determination• Ex#2: y = 80-10x, R2=.49: R = -.70• NOTE! Ex#2 has stronger relationship, as

measured by coefficient of determination

Extreme Values

• R=+1: perfect positive correlation

• R= -1: perfect negative correlation

• R=0: zero correlation

Top Ten #5

• Expected Value = E(x) = ΣxP(x)

= x1P(x1) + x2P(x2) +…

Expected value is a weighted average, also a long-run average

E(x) Example

• Find the expected age at high school graduation if 11 were 17 years old, 80 were 18, and 5 were 19

• Step 1: 11+80+5=96

Step 2

x P(x) xP(x)

17 11/96=.115 17(.115)=1.955

18 80/96=.833 18(.833)=14.994

19 5/96=.052 19(.052)=.988

E(x)= 17.937

Top Ten #6

• What distribution to use?

Use binomial distribution if:

• Random variable (x) is number of successes in n trials

• Each trial is success or failure• Independent trials• Constant probability of success (π) on each trial• Sampling with replacement (in practice, people

may use binomial w/o replacement, but theory is with replacement)

Success vs failure

• Male vs female

• Defective vs ok

• Yes or no

• Pass (8 or more right answers) vs fail (fewer than 8)

• Buy drink (21 or over) vs can’t buy drink

Binomial is discrete

• Integer values

• 0,1,2,…n

• Binomial is often skewed, but may be symmetric

Normal Distribution

• Continuous, bell-shaped, symmetric• Mean=median=mode• Measurement (dollars, inches, years)• Cumulative probability under normal curve : use Z

table if you know population mean and population standard deviation

• Sample mean: use Z table if you know population standard deviation and either normal population or n > 30

t distribution

• Continuous, bell-shaped, symmetric• Applications similar to normal• More spread out than normal• Use t if normal population but population

standard deviation not known• Degrees of freedom = df = n-1 if estimating

the mean of one population • t approaches z as df increases

Top Ten #7

• P-value = probability of getting a sample statistic as extreme (or more extreme) than the sample statistic you got from your sample

P-value example: 1 tail test

• Ho: µ = 40

• HA: µ > 40

• Sample mean = 43

• P-value = P(sample mean > 43, given Ho true)

• Reject Ho if p-value < α (significance level)

Two cases

• Suppose α = .05

• Case 1: p-value = .02, then reject Ho (unlikely Ho is true; you believe population mean > 40)

• Case 2: p-value = .08, then do not reject Ho (Ho may be true; you have reason to believe that population mean may be 40)

P-value example: 2 tail test

• Ho: µ = 70

• HA: µ not equal to 70

• Sample mean = 72

• If 2-tails, then P-value =

2*P(sample mean > 72)=2(.04)=.08

If α = .05, p-value > α, so do not reject Ho

Top Ten #8

• Variation creates uncertainty

No variation

• Certainty, exact prediction

• Standard deviation = 0

• Variance = 0

• All data exactly same

• Example: all workers in minimum wage job

High variation

• Uncertainty, unpredictable

• High standard deviation

• Ex#1: Workers in downtown L.A. have variation between CEOs and garment workers

• Ex#2: New York temperatures in spring range from below freezing to very hot

Comparing standard deviations

• Temperature Example

• Beach city: small standard deviation (single temperature reading close to mean)

• High Desert city: High standard deviation (hot days, cool nights in spring)

Standard error of the mean

• Standard deviation of sample mean =

standard deviation/square root of n

Ex: standard deviation = 10, n =4, so standard error of the mean = 10/2= 5

Note that 5<10, so standard error < standard deviation

As n increases, standard error decreases

Sampling Distribution

• Expected value of sample mean = population mean, but an individual sample mean could be smaller or larger than the population mean

• Population mean is a constant parameter, but sample mean is a random variable

• Sampling distribution is distribution of sample means

Example

• Mean age of all students in the building is population mean

• Each classroom has a sample mean

• Distribution of sample means from all classrooms is sampling distribution

Central Limit Theorem

• If population standard deviation is known, sampling distribution of sample means is normal if n > 30

• CLT applies even if original population is skewed

Top Ten #9

• Population vs sample

Population

• Collection of all items(all light bulbs made at factory)

• Parameter: measure of population

(1)population mean(average number of hours in life of all bulbs)

(2)population proportion(% of all bulbs that are defective)

Sample

• Part of population(bulbs tested by inspector)

• Statistic: measure of sample = estimate of parameter

(1) sample mean(average number of hours in life of bulbs tested by inspector)

(2) sample proportion(% of bulbs in sample that are defective)

Top Ten #10

• Qualitative vs quantitative

Qualitative

• Categorical data

success vs failure

ethnicity

marital status

color

zip code

4 star hotel in tour guide

Qualitative

• If you need an “average”, do not calculate the mean

• However, you can compute the mode (“average” person is married, buys a blue car made in America)

Quantitative

• 2 cases

• Case 1: discrete

• Case 2: continuous

Discrete

(1) integer values (0,1,2,…)

(2) example: binomial

(3) finite number of possible values

(4) counting

(5) number of brothers

(6) number of cars arriving at gas station

Continuous

• Real numbers, such as decimal values ($22.22)

• Examples: Z, t

• Infinite number of possible values

• Measurement

• Miles per gallon, distance, duration of time

Graphical tools

• Pie chart or bar chart: qualitative

• Joint frequency table: qualitative (relate marital status vs zip code)

• Scatter diagram: quantitative (distance from CSUN vs duration of time to reach CSUN)

Hypothesis testingConfidence intervals

• Quantitative: Mean

• Qualitative: Proportion