The attached data set that gives the distance of the longest drop (in feet) and

Question

The attached data set that gives the distance of the longest drop (in feet) and the top speed (in mph) of the top roller coasters across the country should be used to answer all parts of this question.

Name Speed Drop

Millennium Force 99 350

Goliath 85 255

Titan 85 255

Phantom's Revenge 82 228

Xcelerator 62 120

Desperado 80 225

HyperSonic XLC 80 133

Nitro 80 215

Phantom's Revenge 80 225

Son Of Beast 78.4 214

Superman - Ride Of Steel 79 225

X 76 215

Mamba 75 205

Steel Force 75 205

Wild Thing 74 196

Apollo's Chariot 73 210

Raging Bull 73 208

Rattler 73 166.5

Superman - Ride Of Steel 73 205

Magnum XL 79 209

Viper 70 171

The Blast Coaster 70 80

Great American Scream Machine 68 155

Alpengeist 67 170

Incredible Hulk 67 105

Manhattan Express 67 144

Boss 66.3 150

American Eagle 66 147

Wildfire 66 155

Deja Vu 65.6 177

Batman Knight Flight 65 148

Hercules 65 151

Kraken 65 144

Mean Streak 65 155

Medusa 65 150

Orient Express 65 115

Rattler 65 124

Riddler's Revenge 65 146

Steel Eel 65 150

Texas Giant 67 151

Beast 64.8 141

Chang 63 144

Scream! 63 141

Tennessee Tornado 63 128

Colossus 62 115

Screamin' Eagle 62 92

Hangman 55 95

Hurricane 55 100

Invertigo 61 138

Iron Wolf 55 90

Kong 55 95

Mind Eraser 55 95

Silver Bullet 55 75

Starliner 55 76

T2 55 95

Thunderbolt 55 95

Timber Wolf 53 95

Wild One 53 88

Cheetah 52 90

Cannon Ball 50 70

Coaster Thrill Ride 50 52

Comet 50 78

Mexico Rattler 47 75

Corkscrew 46 62

Afterburner 45 47

Whizzer 41 67

Canyon Blaster 41 66

Blue Streak 40 72

Steel Dragon 95 306.75

Thunder Dolphin 80.8 218.1

Titan 71.5 178

Oblivion 68 180

Stunt Fall 65.6 177

Hayabusa 60.3 124.67

Top Gun 56 93

(I wish I could attach the statcrunch/excel sheet here. Hoping this is not too difficult to understand. Thank you for your time!)

Part 1: Describe the association in context. Remember this has 4 parts! Talk about strength, direction, form and what the relationship means in context.

Part 2: What is the r2 value and what does it mean in the context of the question?

Part 3: Write the equation of the least squares regression line in context. Be sure to use the equation editor. You can find the "hat" button on the 7th tab in the 3rd section, as shown below. Write your y-intercept to one decimal place and your slope to 3 decimal places.

Part 4: Explain (in context) what the slope of the line means.

Part 5: Explain (in context) what the y-intercept of the line means.

Part 6: Using your model from question 5, the top speed of a roller coaster that has a drop of 300 feet is _______ mph. Round your result to one decimal place.

Explanation / Answer

Answer:

Here, since the data is a quantitative data, the proper measure of association between two type of variable will be correlation. Correlation measures the linear association
between two sets of data. One important aspect about correlation must be taken under consideration while using it is, Correlation does not implies any form of casual relationship between two
variable. It is a simple measure of linear association.

Here correlation coefficient is : 0.9333429

In context of linear regression r-square statistics is nothing but the square of this correlation coefficient. This is a measure of quality of goodness of fit if the other
assumption of regression holds true. It also means the amount of variance explained by the dependent variable by the independent variable.

For this model: Drop= Intercept+ slope*Speed

R-squared: 0.8711,   Adjusted R-squared: 0.8694

Fitted model is : Speed = 38.322453 + 0.182712 *Drop

Intercept: It means that the average value of Drop will be 38.322453 if all the other value of Speed is zero.

Slope: It means that for per unit change in speed drop will be change by 0.182712 unit.

93.13609 will be predicted value of speed for a given drop of 300.

Rcode:

> x<-c(99,350,85,255,85,255,82,228,62,120,80,225,80,133,80,215,80,225,78.4,214,79,225,76,215,75,205,75,205,74,196,73,210,73,208,73,166.5,73,205,79,209,70,171,70,80,68,155,67,170,67,105,67,144,66.3,150,66,147,66,155,65.6,177,65,148,65,151,65,144,65,155,65,150,65,115,65,124,65,146,65,150,67,151,64.8,141,63,144,63,141,63,128,62,115,62,92,55,95,55,100,61,138,55,90,55,95,55,95,55,75,55,76,55,95,55,95,53,95,53,88,52,90,50,70,50,52,50,78,47,75,46,62,45,47,41,67,41,66,40,72,95,306.75,80.8,218.1,71.5,178,68,180,65.6,177,60.3,124.67,56,93)
> df<-data.frame(matrix(x,ncol=2,byrow = T))
> names(df)<-c("Speed","Drop")
> cor(df)
Speed Drop
Speed 1.0000000 0.9333429
Drop 0.9333429 1.0000000
> model<-lm(Speed~Drop, data=df)
> summary(model)

Call:
lm(formula = Speed ~ Drop, data = df)

Residuals:
Min 1Q Median 3Q Max
-11.4777 -1.6379 -0.4327 1.1892 17.3768

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 38.322453 1.307337 29.31 <2e-16 ***
Drop 0.182712 0.008225 22.21 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.347 on 73 degrees of freedom
Multiple R-squared: 0.8711,   Adjusted R-squared: 0.8694
F-statistic: 493.5 on 1 and 73 DF, p-value: < 2.2e-16

> predict(model,data.frame(Drop=c(300)))
1
93.13609

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