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[Solved] STA304 Assignment 2 Fall 2019  

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admin
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05/02/2020 10:13 pm  

STA304 Applied Statistics Assignment 2 Solution & Discussion Fall 2019


Question                                                                                

For the following time series, determine the trend by using the method of (i) semi-averages, (ii) 3-year moving averages, and (iii) least-squares for fitting a straight line:

Year

1968

1969

1970

1971

1972

1973

1974

1975

1976

Values of series

2

4

6

8

7

6

8

10

2

 

Which of the trend do you prefer, and why?

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admin
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Admin
Joined: 1 year ago
Posts: 2359
05/02/2020 10:18 pm  

STA304 Applied Statistics Assignment 2 Solution & Discussion Fall 2019


Solution Idea:

 

I) semi –average

year

value

Total

Average

Trend

1968

2

 

 

3.8

1969

4

 

 

4.6

 

 

20

5

 

1970

6

 

 

5.4

1971

8

 

 

6.2

1972

7

 

 

7.0

1973

6

 

 

7.8

1974

8

 

 

8.6

 

 

 

 

 

 

 

36

9

 

1975

10

 

 

9.4

1976

12

 

 

10.2

 

ii) 3 year moving:

value

Total

Average

2

--

--

4

12

4.0

 

 

 

6

18

6.0

8

21

7.0

7

21

7.0

6

21

7.0

8

24

8.0

 

 

 

 

 

 

10

30

10.0

12

--

--

 

iii) least-squares for fitting a straight line.

year

value

X

 

XY

Trend value

1968

2

-4

16

-18

3

1969

4

-3

9

-12

4

1970

6

-2

4

-12

5

1971

7

-1

1

-8

6

1972

7

0

0

-40

7

1973

6

1

1

6

8

1974

8

2

4

16

9

1975

10

3

9

30

10

1976

12

4

16

48

11

Total

63

0

60

+100\+60

63

 

Let the equation of the straight line be

 {{Y_t} = a + bX}\limits^ \wedge

Then the two normal equations reduce to

\begin{array}{l}
\sum {X = na} \\
\sum {XY = b\sum {{X^2}} } 
\end{array}

Substituating the value, we get

\begin{array}{l}
a = \frac{{\sum Y }}{n} = \frac{{63}}{9} = 7\\
b = \frac{{\sum {XY} }}{{\sum {{X^2}} }} = \frac{{60}}{{60}} = 1
\end{array}

Hence the require is  {{Y_t} = 7 + X}\limits^ \wedge  with the origin at 1972. The trend values are obtained bu substituating the value of x in the first equation.these values appear in the last column of the above table. We prefer the least square trend as it is the better.

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