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: Create scatter - diagram

On a graph, mark the points \((1014,50)\), \((935,80)\), \((980,60)\), \((955,135)\), \((995,84)\) where the \(x\) - axis represents the barometer reading and the \(y\) - axis represents the maximum wind speed.

Step3: Calculate regression line (least - squares line)

The equation of the least - squares line is \(y = a+bx\), 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 \(a=\bar{y}-b\bar{x}\), \(n = 5\), \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\)

First, calculate \(\sum_{i = 1}^{5}x_i=1014 + 935+980+955+995=4879\), \(\sum_{i = 1}^{5}y_i=50 + 80+60+135+84=409\), \(\sum_{i = 1}^{5}x_i^{2}=1014^{2}+935^{2}+980^{2}+955^{2}+995^{2}=4879087\), \(\sum_{i = 1}^{5}x_iy_i=1014\times50+935\times80+980\times60+955\times135+995\times84 = 437715\)

\(\bar{x}=\frac{4879}{5}=975.8\), \(\bar{y}=\frac{409}{5}=81.8\)

\(b=\frac{5\times437715 - 4879\times409}{5\times4879087-(4879)^{2}}\)
\[

$$\begin{align*} b&=\frac{2188575-1995511}{24395435 - 23804641}\\ &=\frac{193064}{590794}\\ &\approx - 0.327 \end{align*}$$

\]

\(a=81.8-(- 0.327)\times975.8=81.8 + 319.1866=400.9866\)

The equation of the least - squares line is \(y = 400.99-0.33x\)

Then draw the line \(y = 400.99-0.33x\) on the scatter - diagram.

This is a manual way to find the line. In practice, graphing calculators or software like Excel, Python (using libraries like matplotlib and numpy) can be used to create the scatter - diagram and draw the least - squares line more accurately.

Answer:

To answer this question fully, one needs to create a scatter - diagram with points \((1014,50)\), \((935,80)\), \((980,60)\), \((955,135)\), \((995,84)\) and draw the line \(y = 400.99-0.33x\) on it. The actual graph cannot be provided in this text - based format, but the steps to create it are as above.