Question

Example # 4: Some people love roller coasters. Prof. Bodrero found a site with info on hundreds of coasters in the US. He wonders if the height (in meters) of the roller coaster can predict the max speed (km/hour). The info for a random sample of 10 roller coasters is below. Source: statcrunch.com Height (m) 58 13 27 57 30 44 3 0 26 24 49 Speed (km/hr) 105 62 81 113 96 100 81 78 83 105 a). Construct a scatter plot for the data. (Hint: first determine which data is x and which is y). b). Calculater and then interpretr? c). Test the data for linear correlation. Use a = 0.05. d). Find the regression equation and then graph it on the scatter plot of the data. e). If a roller coaster is 40 meters high, what does the model predict for maximum speed? f). Find and interpret the marginal change (in y as x changes by one unit).

Answer 1

a)

Scatter plot 0 10 20 30 40 50 60 70

where x= height of roller coaster

y = speed of roller coaster

b) The correlation coefficient is given by

$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^{2}-(\sum x)^{2}}\sqrt{n\sum y^{2}-(\sum y)^{2}}}$

Calculation

	x	y	x^2	y^2	xy
	58	105	3364	11025	6090
	13	62	169	3844	806
	27	81	729	6561	2187
	57	113	3249	12769	6441
	30	96	900	9216	2880
	44	100	1936	10000	4400
	30	81	900	6561	2430
	26	78	676	6084	2028
	24	83	576	6889	1992
	49	105	2401	11025	5145
sum	358	904	14900	83974	34399

Thus

$r=\frac{10*34399-358*904}{\sqrt{10*14900-(358)^{2}}\sqrt{10*83974-(904)^{2}}}$

= 0.9397

$r^2=0.8831$

Interpretation : r^2 is called coefficient of determination which is interpreted as , 88.31% change in y variable (dependent variable) can be explained by x variable(independent variable) .

c) The null and alternative hypothesis

$H0:\rho =0$

$Ha:\rho \neq 0$

Test statistic

? = لا 2 - rvnے

= 7.77

degrees of freedom = n-2 = 8

For , with df = 8 , critical value of t is

9= 0,05

tc = 2.31 ( from ta table )

Since t > 2.31

Reject H0

there is sufficient evidence to conclude that linear correlation exists between height and speed of roller coaster .

d) The equation of line of regression is

$y=\alpha +\beta x$

where estimates of $\alpha ,\beta$ are

$a= \frac{\sum y\sum x^{2}-\sum x\sum xy}{n\sum x^{2}-(\sum x)^{2}}=55.42$

$b= \frac{n\sum xy -\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}=0.98$

Thus ,estimated line of regression is given by

$\hat{y}=55.42+0.98x$

e) x=40 meter

predicted maximum speed is

km /hr

= 55.42+0.98 + 40 = 94.62

f) Slope is =0.98

Thus for each meter increase in x , y is expected to increase by 0.98 km/hr

d) The graph of regression equation in scatter plot

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