ST 350 ReilandLECTURE WORKSHEET #32

SOLUTIONS NAME

Inference for the Least Squares Line

Since 1994 the Best Roller Coaster Poll(www.ushsho.com/bestrollercoasterpoll.com) has been ranking the world's best rollercoasters. Shown below are the maximum speed (mph) and height (ft) of the top tenroller coasters from a recent poll.

Coaster Location Height (ft) Max Speed (mph)B CBizarro MA 208 77

Expedition GeForce DE 188 74.6Intimidator 305 VA 305 93

Kawasemi JP 108 54Nemesis UK 43 50Piraten DK 105 56Goliath NL 155 66.9

Millenium Force OH 310 93Katun IT 169 65iSpeed IT 180 74.6

A scatterplot of the speed against height is shown at the top of the next page.

ST 350 LECTURE WORKSHEET #32 page 2

1. B œ "((Þ" C œ (!Þ%" = œ )%Þ!(# = œ "&Þ!$% < œ !Þ*)$#B C

slope , œ < œ !Þ*)$# œ Þ"(&)"== )%Þ!(#

"&Þ!$%C

B

intercept , œ C , B œ (!Þ%" Þ"(&)Ð"((Þ"Ñ œ $*Þ#(! "

least squares prediction line is Cs œ $*Þ#( !Þ"(&)B

2. For the purposes of doing statistical inference in simple linear regression, summarize theassumptions that are made when using the linear model

C œ B 3 ! " 3" " %3

to model the relationship between and .B C 1. The distribution of errors can be modeled by a Normal model 2. IÐ Ñ œ !%3 3. for all (that is, the standard deviation of the errors is constant forWHÐ Ñ œ B% 53 %

all )B 4. The errors are independent.%3ß "ß #ß ÞÞÞß 8

In summary, ( stands for independent and identically% 53µ 33.RÐ!ß Ñ 33.%

distributed).

3. Calculate the estimate of the standard deviation of the error term.=% %5 : Note WWI œ '(Þ))&"''

= œ œ% WWI8#

'(Þ)*) ¸ #Þ*"

ST 350 LECTURE WORKSHEET #32 page 3

4. Is the slope significantly different from 0? (that is, is the explanatory variable "" Heightuseful for predicting the ?).Maximum Speed

L À œ !! "" L À Á !+ ""

WIÐ, Ñ œ = œ" ,"=

8" =%

B

œ ¸ Þ!""&#Þ*"

* )%Þ!(# Test Statistic: > œ œ œ "&Þ##, !

= !Þ!""&!Þ"(&)"

,"

-valueT œ TÐ> Ÿ l Ñ TÐ>   l Ñ) )test statistic| test statistic|œ !Þ!!!!!!"( !Þ!!!!!!"( œ !Þ!!!!!!$% (see picture below)

CONCLUSION: Since the is very, very small, there is strong evidenceT @+6?/against the null hypothesis. Therefore, we reject the null hypothesis inL À œ !! ""favor of the alternative hypothesis and conclude that the slope differsL À Á !+ ""from 0. In the context of the problem, we conclude that the explanatory variableHeight Maximum Speed is useful for predicting the .

5. What is a 95% confidence interval for the slope ?"" For 95% confidence, > œ #Þ$!'!

)

, „ > ñ= œ !Þ"(&) „ #Þ$!'!Ð!Þ!""&Ñ œ !Þ"(&) „ !Þ!#'&# Ê" ) ,"

Ð!Þ"%*#)ß !Þ#!#$#Ñ

6. An amusement park owner wants to build a new roller coaster called the CraniumShaker with 250 feet, what is the predicted ?Height Maximum SpeedB œ

Cs œ , , ñ #&! œ $*Þ#( !Þ"(&)Ð#&!Ñ œ )$Þ##7:2! "

ST 350 LECTURE WORKSHEET #32 page 4

7. Calculate a 95% interval for the (recallprediction Cranium Shaker's Maximum Speedthat the height is 250 feet)..

Cs „ > ñ = ñÐ#&! BÑ = œ‡ #,

# #=8 /

"

/#

)$Þ## „ #Þ$!'! Ð!Þ!""&Ñ Ð#&! "((Þ"Ñ #Þ*" # # ##Þ*""!

#

œ )$Þ## „ #Þ$!'!Ð$Þ"''Ñ œ )$Þ## „ (Þ$ Ê Ð(&Þ*#ß *!Þ&#Ñ

8. Calculate a 95% interval for the mean of all roller coastersconfidence Maximum Speedwith 250 feet.Height B œ

C „ > ñ = ñÐ#&! BÑ œs ‡ #,

# =8

"

/#

)$Þ## „ #Þ$!'! Ð!Þ!""&Ñ Ð#&! "((Þ"Ñ # # #Þ*""!

#

)$Þ## „ #Þ$!'!Ð"Þ&&Ñ œ )$Þ## „ $Þ&( Ê Ð(*Þ'&ß )'Þ(*Ñ