Review of Top 10 Conceptsin Statistics

NOTE: This Power Point file is not an introduction, but rather a checklist of topics to review

Top Ten #1

Descriptive Statistics

Measures of Central Location

Mean Median Mode

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

4 Algebra: Σx = N*µ = 3*4 =12 Sample mean = x-bar = Σx/n Example: the number of hours spent on the

Internet: 4, 8, and 9 x-bar = (4+8+9)/3 = 7 hours Do NOT use if the number of observations is

small or with extreme values Ex: Do NOT use if 3 houses were sold this week,

and one was a mansion

Median Median = middle value Example: 5,1,6

Step 1: Sort data: 1,5,6 Step 2: Middle value = 5

When there is an even number of observation, median is computed by averaging the two observations in the middle.

OK even if there are extreme values Home sales: 100K,200K,900K, so

mean =400K, but median = 200K

Mode Mode: most frequent value Ex: female, male, female

Mode = female Ex: 1,1,2,3,5,8

Mode = 1 It may not be a very good measure, see the

following example

Measures of Central Location - Example

Sample: 0, 0, 5, 7, 8, 9, 12, 14, 22, 23

Sample Mean = x-bar = Σx/n = 100/10 = 10 Median = (8+9)/2 = 8.5 Mode = 0

Relationship

Case 1: if probability distribution symmetric (ex. bell-shaped, normal distribution), Mean = Median = Mode

Case 2: if distribution positively skewed to right (ex. incomes of employers in large firm: a large number of relatively low-paid workers and a small number of high-paid executives), Mode < Median < Mean

Relationship – cont’d

Case 3: if distribution negatively skewed to left (ex. The time taken by students to write exams: few students hand their exams early and majority of students turn in their exam at the end of exam), Mean < Median < Mode

Dispersion – Measures of Variability

How much spread of data How much uncertainty Measures

Range Variance Standard deviation

Range

Range = Max-Min > 0 But range affected by unusual values Ex: Santa Monica has a high of 105 degrees

and a low of 30 once a century, but range would be 105-30 = 75

Standard Deviation (SD)

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

Standard Deviation = Square Root of Variance

1

)( 2

n

xxs

Sample Standard Deviationx

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

Measures of Variability - Example

The hourly wages earned by a sample of five students are:

$7, $5, $11, $8, and $6

Range: 11 – 5 = 6

Variance:

Standard deviation:

30.5

15

2.21

15

4.76...4.77

1

222

2

n

XXs

30.230.52 ss

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

Population mean=µ Population proportion=π A statement about the value of a population

parameter Never include sample statistic (such as, x-

bar) in hypothesis

H0: Null Hypothesis

HA or H1: 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)

One-Tailed Tests

A test is one-tailed when the alternate hypothesis, H1 or HA, states a direction, such as:

• H1: The mean yearly salaries earned by full-time employees is more than $45,000. (µ>$45,000)

• H1: The average speed of cars traveling on freeway is less than 75 miles per hour. (µ<75)

• H1: Less than 20 percent of the customers pay cash for their gasoline purchase. (π <0.2)

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)

Two-Tailed Tests

A test is two-tailed when no direction is specified in the alternate hypothesis

• H1: The mean amount of time spent for the Internet is not equal to 5 hours. (µ 5).

• H1: The mean price for a gallon of gasoline is not equal to $2.54. (µ ≠ $2.54).

Reject Null Hypothesis (H0) If

Absolute value of test statistic* > critical value* Reject H0 if |Z Value| > critical Z

Reject H0 if | t Value| > critical t

Reject H0 if p-value < significance level (alpha) Note that direction of inequality is reversed!

Reject H0 if very large difference between sample statistic and population parameter in H0

* Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

* Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

Example: Smog Check

H0 : µ = 80

HA: µ > 80 If test statistic =2.2 and critical value = 1.96,

reject H0, and conclude that the population mean is likely > 80

If test statistic = 1.6 and critical value = 1.96, do not reject H0, and reserve judgment about H0

Type I vs Type II Error

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

Beta= β = P(type II error) = probability you do not reject a null hypothesis, given H0 false

Ex: H0 : Defendant innocent α = P(jury convicts innocent person) β =P(jury acquits guilty person)

Type I vs Type II Error

H0 true H0 false

Reject H0 Alpha =α =

P(type I error)

1 – β (Correct Decision)

Do not reject H0 1 – α (Correct Decision)

Beta =β =

P(type II error)

Example: Smog Check

H0 : µ = 80

HA: µ > 80

If p-value = 0.01 and alpha = 0.05, reject H0, and conclude that the population mean is likely > 80

If p-value = 0.07 and alpha = 0.05, do not reject H0, and reserve judgment about H0

Test Statistic

When testing for the population mean from a large sample and the population standard deviation is known, the test statistic is given by:

zX

/ n

The processors of Best Mayo indicate on the label that the bottle contains 16 ounces of mayo. The standard deviation of the process is 0.5 ounces. A sample of 36 bottles from last hour’s production showed a mean weight of 16.12 ounces per bottle. At the .05 significance level, can we conclude that the mean amount per bottle is greater than 16 ounces?

Example

1. State the null and the alternative hypotheses:

H0: μ = 16, H1: μ > 16

3. Identify the test statistic. Because we know the population standard deviation, the test statistic is z.

4. State the decision rule.

Reject H0 if |z|> 1.645 (= z0.05)

2. Select the level of significance. In this case, we selected the .05 significance level.

Example – cont’d

5. Compute the value of the test statistic

44.1365.0

00.1612.16

n

Xz

6. Conclusion: Do not reject the null hypothesis. We cannot conclude the mean is greater than 16 ounces.

Example – cont’d

Top Ten #3

Confidence Intervals: Mean and Proportion

Confidence Interval

A confidence interval is a range of values within which the population parameter is expected to occur.

Factors for Confidence Interval

The factors that determine the width of a confidence interval are:

1. The sample size, n2. The variability in the population, usually

estimated by standard deviation.3. The desired level of confidence.

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

nz

n

x

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

9.44 < µ < 10.56

56.01049

296.1

49

490

One of SOM professors wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours. It is assumed that the population standard deviation is 4 hours. What is the population mean?

Another Example

95 percent confidence interval for the population mean.

12.100.2449

496.100.2496.1

n

X

The confidence limits range from 22.88 to 25.12. We estimate with 95 percent confidence that the average number of hours worked per week by students lies between these two values.

Another Example – cont’d

Confidence 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

n

s

n

xtn 1

Confidence Interval: Mean t distribution

Confidence Interval: Proportion Use if success or failure

(ex: defective or not-defective, satisfactory or unsatisfactory)

Normal approximation to binomial ok if (n)(π) > 5 and (n)(1-π) > 5, wheren = sample sizeπ= population proportionNOTE: NEVER use the t table if proportion!!

Confidence Interval: Proportion

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

n = 100, 95% confidence

n

ppzp

)1(

05.08. 100

)92)(.08.0(96.108.

Confidence Interval: Proportion

A sample of 500 people who own their house revealed that 175 planned to sell their homes within five years. Develop a 98% confidence interval for the proportion of people who plan to sell their house within five years.

0497.35. 500

)65)(.35(.33.235.

35.0500

175p

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. In fact, the population mean is a fixed but unknown quantity.)

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 Confidence Interval If

(1) small sample size(n)

(2) large standard deviation

(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

Linear Regression

Regression equation: =dependent variable=predicted value x= independent variable b0=y-intercept =predicted value of y if x=0

b1=slope=regression coefficient

=change in y per unit change in x

xy bb 10ˆ

y

Slope vs Correlation

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

Negative slope (b1<0): negative correlation (y decrease if x increase)

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

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

Slope (b1) = -6.5

Method of Least Squares formulas not on BUS 302 exam

b1= -6.5 given

Interpretation: If one female manager has 1 more child than another, salary is $6,500 lower; that is, salary of female managers is expected to decrease by -6.5 (in thousand of dollars) per child

Intercept (b0)

33.23

7

n

xx 33.44

3

133

n

yy

b0 = 44.33 – (-6.5)(2.33) = 59.5

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

xy bb 10

Regression Equation

xy 5.65.59ˆ

Forecast Salary If 3 Children

59.5 –6.5(3) = 40

$40,000 = expected salary

xforecasty bb 10ˆ

yyerror ˆ

2

)ˆ(

2

2

n

yy

n

SSES

Standard Error of Estimate

Standard Error of Estimate

(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

Standard Error of Estimate

Actual salary typically $1,900 away from expected salary

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

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

Interpretation: 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 (b1) < 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

Summary: 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

MS Excel Output

Correlation Coefficient (-0.9912): Note that you need to change the sign because the sign of slope (b1) is negative (-6.5)

Coefficient of Determination

Standard Error of Estimate

Regression Coefficient

Top Ten #5

Expected Value

Expected Value

Expected Value = E(x) = ΣxP(x)

= x1P(x1) + x2P(x2) +…

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

Example

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

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

The binomial experiment can result in only one of two possible outcomes:

Male vs. Female Defective vs. Non-defective Yes or No Pass (8 or more right answers) vs. Fail (fewer

than 8) Buy drink (21 or over) vs. Cannot 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, mound-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

Normal or t Distribution?

Use t table if normal population but population standard deviation (σ) is not known

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

Top Ten #7

P-value

P-value

P-value = probability of getting a sample statistic as extreme (or more extreme) than the sample statistic you got from your sample, given that the null hypothesis is true

P-value Example: one tail test

H0: µ = 40

HA: µ > 40 Sample mean = 43 P-value = P(sample mean > 43, given H0 true) Meaning: probability of observing a sample mean

as large as 43 when the population mean is 40 How to use it: Reject H0 if p-value < α

(significance level)

Two Cases

Suppose α = .05 Case 1: suppose p-value = .02, then reject H0

(unlikely H0 is true; you believe population mean > 40)

Case 2: suppose p-value = .08, then do not reject H0 (H0 may be true; you have reason to believe that the population mean may be 40)

P-value Example: two tail test

H0 : µ = 70

HA: µ ≠ 70 Sample mean = 72 If two-tails, then P-value =

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

If α = .05, p-value > α, so do not reject H0

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 (CLT)

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

Two 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