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





Review Chapter 2Bar Chart

Pie Chart

Qualitative DataQualitative Data

Histogram

Polygon

Cumulative Frequency

Distribution

Quantitative DataQuantitative Data

•Review





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





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





•Review





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





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





Least Square Regression Equation Ŷ = a + bX

Review Chapter 12

•Review





•Review





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





• Review





Chapter 16: Time Series & Forecasting

• Review





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



Components of a Time Series

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

than one year.

• Review




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

patterns tend to repeat themselves each year.

• Review




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

unidentifiable

• Review




• Useful in smoothing time series to see its trend

• Basic method used in measuring seasonal fluctuation

Moving Average:

• Review





• Review





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





Moving Average:

• Review





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

• Review





• 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





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





• Review





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





• 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





• Review





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





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

Technology

Lesson05