Lesson05

IBS Statistics Year 1

Dr. Ning DING [email protected]

What we are going to learn?

• Review

• Chapter 16: – Define the components of a time series– Compute a moving average– Determine a linear trend equation

Chapter 12: Sim Reg & Corr

Exercise

Ŷ = -1.8182 + 0.1329XŶ = -1.8182 + 0.1329X Sample Exam P.4

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

a

r2

SD

Ŷ = -1.8182 + 0.1329XŶ = -1.8182 + 0.1329X

98.87%

increase

Review Chapter 1

Nominal: gender Ordinal: ranking Interval: temperature, IQ Ratio: age in years

Qualitative:

gender, eye color,

Quantitative:

income, distance

Discrete counting or Continuous measuringDiscrete counting or Continuous measuring

true zero

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Review Chapter 2Bar Chart

Pie Chart

Qualitative DataQualitative Data

Histogram

Polygon

Cumulative Frequency

Distribution

Quantitative DataQuantitative Data

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Review Chapter 3

MeanMean

Median

Ungrouped Data Grouped Data

1 2 2 3 4

(1+2+2+3+4)/5

Classes f 10 up to 20 220 up to 30 130 up to 40 4

2 *15=301 *25=254 *35=140

2 *15=301 *25=254 *35=140

10-<20 220-<30 330-<40 7

L=(N+1)/2(7+1)/2=4

30 40

3 74 5 6

32.5

10-<20 220 -<30 130-<40 4

1957 =24.86ModeMode

Central TendencyCentral Tendency

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Ungrouped Data

1 2 2 3 410 up to 20 220 up to 30 130 up to 40 4

DispersionDispersion

RangeRange

VarianceVariance

Standard DeviationStandard Deviation

4-1=3

Nμ)-Σ(X

=σ2

2

1-n)X-Σ(X

=s2

2

2σ=σ 2s=s

Grouped Data

1957 =24.86

Review Chapter 3

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

The distribution is skewed to __________because the mean is __________the median.

P42 Example Ch2

the right larger than

Interquartile Range

Mean =23.06

Review Chapter 4

Review Chapter 12

Strong & Positivecorrelation

Strong & Negativecorrelation

Moderate & Positivecorrelation

Moderate & Negativecorrelation

Nocorrelation

r = 0.9 r = -0.9 r = 0.6 r = -0.6r = 0

Ŷ = a + bX

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Standard Error

X Axis: Independent Variable

Y A

xis: Dependent V

ariable

r2 = 1 r = -1Standard Error = 0

r2 = 0.24 r = -0.49Standard Error = 2.5

Standard Error

1 2 3 4 5 6 7

7

6

5

4

3.

2

1

Individual values are more scattered from the regression line.

Ŷ = a + bX

Review Chapter 12

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Least Square Regression Equation Ŷ = a + bX

Review Chapter 12

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Sample Exam P.4

Ŷ = -1.8182 + 0.1329XŶ = -1.8182 + 0.1329X

The graph is positive.

There is a strong determination.

r =0.8619 so 86.19% of the variation in Y is explained by the variation in X.

It is a srong positive correlation.

r2=1.05 r = √ 1.05 = 1.025 = $1.025

X

X

X

X

X

Review Chapter 12

•Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Chapter 16: Time Series & Forecasting

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Components of a Time Series

Time series:a collection of data recorded over a period of time (weekly, monthly, quarterly),

an analysis of history, that can be used by management to make current decisions and plans based on long-term forecasting.

– Secular Trend

– Linear– Nonlinear

– Cyclical variation– Rises and Falls over periods longer than one year

– Seasonal variation– Patterns of change within a year, typically repeating themselves

– Residual variation

S

C

S

R

Secular Trend:The smooth long-term direction of a time series.

• Review

•Chapter 16: –Define the components of a time series

–Compute a moving average

–Determine a linear trend equation

Components of a Time Series

Cyclical Variation:The rise and fall of a time series over periods longer

than one year.

• Review

•Chapter 16: –Define the components of a time series

–Compute a moving average

–Determine a linear trend equation

Seasonal Variation:Patterns of change in a time series within a year. These

patterns tend to repeat themselves each year.

• Review

•Chapter 16: –Define the components of a time series

–Compute a moving average

–Determine a linear trend equation

Irregular Variation:• Episodic – unpredictable but identifiable• Residual – also called chance fluctuation and

unidentifiable

• Review

•Chapter 16: –Define the components of a time series

–Compute a moving average

–Determine a linear trend equation

• Useful in smoothing time series to see its trend

• Basic method used in measuring seasonal fluctuation

Moving Average:

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

1+2+3+4+5+4+3=22 / 7 = 3.143

Seven-Year Moving Total Moving Average

2+3+4+5+4+3+2=23 / 7 = 3.2863+4+5+4+3+2+3=24 / 7 = 3.429

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Moving Average:

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Linear TrendThe long term trend of many business series often approximates a straight line

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• Use the least squares method in Simple Linear Regression (Chapter 12) to find the best linear relationship between 2 variables

• Code time (t) and use it as the independent variable

• E.g. let t be 1 for the first year, 2 for the second, and so on (if data are annual)

Ŷ = a + bX Ŷ = a + btŶ = a + bt

Linear Trend

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Linear Trend• Code time (t) and use it as the independent variable• E.g. let t be 1 for the first year, 2 for the second, and

so on (if data are annual)

YearSales

($ mil.)

2005 7

2006 10

2007 9

2008 11

2009 13

Example:The sales of Jensen Foods, a small grocery chain located

in southwest Texas, since 2005 are:

Year tSales

($ mil.)

2005 1 7

2006 2 10

2007 3 9

2008 4 11

2009 5 13

Ŷ = a + btŶ = a + btxn-x

y xn-xy=b

22 a = Y - bX

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Year tSales

($ mil.)

2005 1 7

2006 2 10

2007 3 9

2008 4 11

2009 5 13

Ŷ = 6.1 + 1.3tŶ = 6.1 + 1.3t9*555

10*3*5163

-

-=b = 1.3

a = 10 -1.3*3 = 6.1

Ŷ = a + btŶ = a + btxn-x

y xn-xy=b

22 a = Y - bX

Linear Trend

Step 1 Step 2

Step 3

Step 4

Step 5

Example:The sales of Jensen Foods, a small grocery chain located

in southwest Texas, since 2005 are:

Ŷ = 6.1 + 1.3tŶ = 6.1 + 1.3t

? 2011 ? Ŷ = 6.1 + 1.3*7 = 15.2Ŷ = 6.1 + 1.3*7 = 15.2$ millions

Linear Trend

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• The amounts spent in vending machines in the United States, in billions of dollars, for the years 1999 through 2005 are given below. Determine the least-squares trend equation and estimate vending sales for 2007.

Exercise

P152 N6 Ch16

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

Ŷ = a + btŶ = a + bt

xn-x

y xn-xy=b

22

a = Y - bX

16*7140

67.22*4*71.683

-

-=b = 1.73

a = 22.67 -1.73*4 = 15.75

P152 N6 Ch16

Exercise

• Review

•Chapter 16: –Define the

components of a time series

–Compute a moving average

–Determine a linear trend equation

P152 N6 Ch16

Ŷ = 15.75 + 1.73tŶ = 15.75 + 1.73t

16*7140

67.22*4*71.683

-

-=b

= 1.73 a = 22.67 -1.73*4 = 15.75

2006 8 Ŷ2007 9

Ŷ = 15.75 + 1.73*9 = $31.32 billions Ŷ = 15.75 + 1.73*9 = $31.32 billions

Exercise

Ŷ = a + btŶ = a + bt xn-x

y xn-xy=b

22

a = Y - bX

P152 N6 Ch16

HintStep 1

Step 2

Step 3

Step 4

Code the year

Calculate X*Y, X2

Step 5

Ŷ = ? + ?*tŶ = ? + ?*t

What is the b?

What is the a?

Formulate the least square equation

What we have learnt?

•Review

•Chapter 16: –Define the components of a time series–Compute a moving average–Determine a linear trend equation