FIRST MIDTERM EXAMINATION SOLUTIONS

FALL, 1999

 

1. A multimedia statistics learning system includes a test of skill in using the computer’s mouse. The software displays a circle at a random location on the computer screen. The subject tries to click in the circle with the mouse as quickly as possible. A new circle appears as soon as the subject clicks the old one. The following is the time required to click in the new circle, in milliseconds for 20 left-handed subjects.

240, 190, 170, 125, 315, 240, 141, 210, 200, 401, 320, 113, 176, 211, 238, 316, 176, 173, 210, 170

a. Find the mean and range for the time. Mean=216.75, Range=401-113=288

b. Construct a stemplot for the the time and interpret.

Most of the observations are around 100’s. The time varies from 113 to 401. The shape is J-shaped, skewed to the right.

c. Give the five-number summary for the time.

Min= 113, First Quartile= 171.5, Median= 205, Third Quartile=240, Max=401

d. Construct a boxplot for the time and interpret.

Centered around 400, varies from 110 to 400. It is skewed to the right.

e. Use the criterion to spot suspected outliers. Please show all of your calculations.

IQR=240-171.5=68.5, 1.5xIQR=102.75

Q3+1.5xIQR=240+102.75=342.75

Q1-1.5xIQR=171.5-102.75=68.75

401 is a suspected outlier.

2. In one high-school graduating class, mean mathematics score on the Scholastic Assessment Test (SAT) was 493 and the standard deviation was 105; the mean verbal score was 420 and the standard deviation was 98. A student in the graduating class scored 703 on the math and 665 on the verbal. Relative to the other students in the graduating class, on which test did the student do better? Please justify your answer.

Mathematics: z-score=(703-493)/105=2

Verbal: z-score=(665-420)/98=2.5

The student did relatively better in verbal.

3. As reported by Runner’s World magazine, the times of the finishers in the New York City 10-km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes.

a. What proportion of finishers in the New York city 10-km run finished the run in less than 45 minutes?

Proportion with x<45 minutes = Proportion with z<-1.78 = 0.0375

b. What proportion of finishers in the New York city 10-km run finished the run in more than 75 minutes?

Proportion with x>75 minutes = Proportion with z>1.56 = 1-0.9406=0.0594

c. What proportion of finishers in the New York city 10-km run finished the run between 50 and 70 minutes?

Proportion with 50<x<70 minutes = Proportion with —1.22<z<1.00 =

0.8413-0.1112=0.7301

d. To be among the 5% of the fastest finishers, what should be the finishing time?

z=-1.645, x=m+sz=61-14.805=46.195. The finishing time should be less than 46.195 minutes,

 

 

 

 

 

 

4. The Kelley Blue Book provides information on whole and retail prices of cars. Following are age and price data for 10 randomly selected Corvettes between 1 and 6 years old. Here x denotes age, in years, and y denotes price, in hundreds of dollars.

 

Car

1

2

3

4

5

6

7

8

9

10

x

6

6

6

2

2

5

4

5

1

4

y

175

165

180

310

269

200

240

213

310

210

a. The least squares regression line is .

Interpret 342 and —28 in the context of the problem.

The average price of a new Corvette is $34,200. On the average for each additional year the car losses $2,800 of its value.

b. Predict the retail price of a 6 years old Corvette.

Predicted value=342-28x6=342-168=174 ($17,400)

c. Find the value of the residual for Car 6.

Observed y=200, predicted y=342-28x5=342-140=202, Residual=200-202=-2

d. The value of the correlation coefficient between age and retail price of the car is —0.97

What is the proportion of variation in retail price of Corvette that can be explained by the age? 0.9409=94.09%

 

5. Interpret the following scatter plots and specify whether the correlation coefficient r is negative, positive or close to zero for each plot.

 

No association, no functional relationship, no apparent outliers. Correlation will be close to zero.

Positive association as amount of drug increases the reaction time increases. Slightly curved functional relation, one outlier at the down-right corner. Correlation will be positive.