Answer to Question #94859 in Statistics and Probability for Octavius

Question

Predict week 11 using the different methods requested and then use historical analysis to determine the best method.

Smoothing:
Week Attendance

1 350
2 500
3 750
4 705
5 820
6 743
7 880
8 900
9 795
10 900 three week four week three week weighted

Next Week Projection (week 11):

Longest Regression:

1) Use regression analysis to predict week 11 in the above example:

Regression can also be used for predictions within the relevant range:

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} Week & Attendance \\ \hline 1 & 350 \\ \hdashline 2 & 500 \\ \hdashline 3 & 750 \\ \hdashline 4 & 705 \\ \hdashline 5 & 820 \\ \hdashline 6 & 743 \\ \hdashline 7 & 880 \\ \hdashline 8 & 900 \\ \hdashline 9 & 795\\ \hdashline 10 & 900 \\ \hline \end{array}$$

$$\sum x_i=55, \sum y_i=7343$$

$$\sum x_i^2=385, \sum y_i^2=5682899$$

$$\sum x_iy_i=44493$$

$$mean:\ \bar{x}={\sum x_i \over n},\ \bar{y}={\sum y_i \over n}$$ $$mean:\ \bar{x}={55\over 10}=5.5,\ \bar{y}={7343 \over 10}=734.3$$ $$Trend \ line: y=A+Bx, B={S_{xy} \over S_{xx}}, A=\bar{y}-B\bar{x}$$ $$correlation\ coefficient: r={S_{xy}\over \sqrt{S_{xx}}\sqrt{S_{yy}}}$$

$$S_{xx}=\sum(x_i-\bar{x})^2=\sum x_i^2-n\cdot\bar{x}^2$$

$$S_{xx}=385-10\cdot5.5^2=82.5$$

$$S_{yy}=\sum(y_i-\bar{y})^2=\sum y_i^2-n\cdot\bar{y}^2$$

$$S_{yy}=5682899-10\cdot734.3^2=290934.1$$

$$S_{xy}=\sum(x_i-\bar{x})(y_i-\bar{y})=\sum x_iy_i-n\cdot\bar{x}\bar{y}$$

$$S_{xy}=44493-10\cdot5.5\cdot734.3=4106.5$$

$$B={S_{xy} \over S_{xx}}={4106.5\over 82.5}\approx49.77575758$$

$$A=\bar{y}-B\bar{x}=734.3-49.77575758\cdot5.5=460.5333333$$

$$r={S_{xy}\over \sqrt{S_{xx}}\sqrt{S_{yy}}}={4106.5\over \sqrt{82.5}\sqrt{290934.1}}\approx0.8382$$

$$y=460.5333333+49.77575758x$$

$$y(11)=460.5333333+49.77575758(11)\approx1008$$