Top Ten 1 Descriptive Statistics NOTE This Power

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: 100 K, 200 K, 900 K, so mean =400 K, but median = 200 K

Location • • Mode: most frequent value Ex: female, 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

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

Standard deviation = Square root of variance

Sample Standard Deviation X 6 6 7 8 13 Sum=40 Mean=40/5=8 6 -8=-2 7 -8=-1 8 -8=0 13 -8=5 Sum=0 (-2)= 4 4 (-1)= 1 0 (5)(5)= 25 Sum = 34

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

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 Reject Ho Ho false 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/n 1/2), where n 1/2 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/n 1/2)

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+b 1 x • y=dependent variable=predicted value • x= independent variable • bo=y-intercept =predicted value of y if x=0 • b 1=slope=regression coefficient =change in y per unit change in x

Slope vs correlation • Positive slope (b 1>0): positive correlation between x and y (y incr if x incr) • Negative slope (b 1<0): negative correlation (y decr if x incr) • Zero slope (b 1=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 • R 2 = % 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 • R 2 = 1: max possible value: perfect linear relationship between y and x (straight line) • R 2 = 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 Sum=7 Sum=133 n=3

Slope = -6. 500 • Method of Least Squares formulas not on 301 exam • B 1 = -6. 500 given

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

Intercept bo= y – b 1 x

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. 5 X

Forecast salary if 3 children 59. 5 – 6. 5(3) = 40 $40, 000 = expected salary

Standard error (1)=x (2)=y 48 (3) (4)= 59. 5 -6. 5 x (2)-(3) 46. 5 1. 5 2 2. 25 1 52 53 -1 1 4 33 33. 5 -. 5 . 25 SSE=3. 5

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 • R 2 = SSR SST • R 2 = 197 =. 98 200. 5 • Interpret: 98% of total variation in salary can be explained by variation in number of children

0< 2 R <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

Our Example • Slope = b 1 = -6. 5 • R 2 =. 98 • R = -. 99

Case 2: Slope > 0 • R is positive square root of coefficient of determination • Ex: R 2 =. 49 • R =. 70 • R has no interpretation • R overstates relationship

Caution • Nonlinear relationship (parabola, hyperbola, etc) can NOT be measured by R 2 • In fact, you could get R 2=0 with a nonlinear graph on a scatter diagram

R=correlation coefficient • Case 1: If b 1>0, R is the positive square root of the coefficient of determination • Ex#1: y = 4+3 x, R 2=. 36: R = +. 60 • Case 2: If b 1<0, R is the negative square root of the coefficient of determination • Ex#2: y = 80 -10 x, R 2=. 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) = Σx. P(x) = x 1 P(x 1) + x 2 P(x 2) +… 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) x. P(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 testing Confidence intervals • Quantitative: Mean • Qualitative: Proportion