Question

can a low barometer reading be used to predict maximum wind spe be the maximum wind speed (in miles per hour) of the cyclone. suppose a random sample of cyclones gave the following information.

x	1014	935	980	955	995
y	50	80	60	135	84

make a scatter diagram for the data. draw the line that best fits the data.

Explanation:

Step1: Prepare data points

We have data points \((x_1,y_1)=(1014,50)\), \((x_2,y_2)=(935,80)\), \((x_3,y_3)=(980,60)\), \((x_4,y_4)=(955,135)\), \((x_5,y_5)=(995,84)\)

Step2: Plot scatter - diagram

On a graph with \(x\) - axis representing barometer reading and \(y\) - axis representing maximum wind speed, plot the above - mentioned points.

Step3: Calculate regression line

The equation of the least - squares regression line is \(\hat{y}=a + bx\), where \(b=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(a=\bar{y}-b\bar{x}\)

First, calculate the necessary sums:
\(n = 5\)
\(\sum_{i=1}^{5}x_i=1014 + 935+980+955+995=4879\)
\(\sum_{i=1}^{5}y_i=50 + 80+60+135+84=409\)
\(\sum_{i=1}^{5}x_i^{2}=1014^{2}+935^{2}+980^{2}+955^{2}+995^{2}\)
\(=1028196+874225+960400+912025+990025 = 4764871\)
\(\sum_{i=1}^{5}x_iy_i=1014\times50+935\times80+980\times60+955\times135+995\times84\)
\(=50700+74800+58800+129925+83580=397805\)

\(b=\frac{5\times397805 - 4879\times409}{5\times4764871-(4879)^{2}}\)
\(=\frac{1989025-1995511}{23824355 - 23804641}=\frac{- 6486}{19714}\approx - 0.33\)

\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{4879}{5}=975.8\)
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{409}{5}=81.8\)

\(a=81.8-(- 0.33)\times975.8=81.8 + 322.014=403.814\)

The regression line is \(\hat{y}=403.814-0.33x\)

Draw this line on the scatter - diagram.

Answer:

The scatter - diagram has been plotted with the points \((1014,50)\), \((935,80)\), \((980,60)\), \((955,135)\), \((995,84)\) and the regression line \(\hat{y}=403.814 - 0.33x\) has been drawn on it.