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Math-Stat 160 Midterm Exam

Tue Thur 2 pm Class

February 20, 2001

Attempt all problems. The following 5 questions refer to the following situation. We wish to predict the mileage per gallon of automobiles based on the level of octane contained in the gasoline used. The following table gives data on the mileage per gallon obtained by a test automobile when using gasoline of varying levels of octane.
Octane (X) Mileage (Y)
89 13.0
93 13.2
87 13.0
90 13.6
89 13.3
95 13.8
100 14.1
98 14.0
Here is the scatter plot of the data.
\epsfig{file=scatter.ps, width=4in, height=5in, angle=-90}
1.
What is the response variable in this study?
(a)
There is no response variable in this study.
(b)
level of octane.
(c)
mileage per gallon and level of octane.
(d)
mileage per gallon.
2.
Guess the correlation coefficient (r) based on the scatter plot.
(a)
0.04
(b)
-0.32
(c)
0.89
(d)
13.6
3.
The least squares procedure gave

\begin{displaymath}\hat{Y} = 5.8 + 0.08 X
\end{displaymath}

Obtain the estimate of the residual when the octane level is 90 (i.e.X = 90).
(a)
More information is needed.
(b)
13.0
(c)
8.6
(d)
0.6
4.
Which of the following is the best description of the intercept of the model given in the previous problem.
(a)
When the level of octane increases by 1, the estimate of mileage increases by 0.08
(b)
When the level of octane is 0, the estimate of mileage is 5.8.
(c)
When the mileage is 0, the level of octane is 5.8.
(d)
It has no practical meaning as X=0 is not in the range of the sample data.

5.
Suppose a student fit both the least squares and Wilcoxon fits for the data above. A second student did the same but with the final mileage incorrectly entered as 140 instead of 14.0. Which result of the two students will differ dramatically?
(a)
Everything will be exactly the same
(b)
the Wilcoxon fit
(c)
the least squares fit.
(d)
Both least squares and Wilcoxon fits will be dramatically different.
6.
The following residual plot was obtained after fitting a linear regression equation. What does the residual plot say about the fit?
\epsfig{file=resplot.ps, width=4in, height= 5in, angle=-90}
(a)
It has no practical meaning as X=0 is not in the rangeM of the sample data.M

(b)
There is an increasing pattern. So, the fit is good.
(c)
There is an increasing pattern. So, the fit is not good.
(d)
The residual plot does not tell us anything.
7.
Which of the following is a true statement concerning probabilities?
(a)
The complement of an impossible event has a probability of zero.
(b)
Negative probabilities are possible if we are absolutely certain that the event cannot occur.
(c)
Even when the probability of an event is one there is still a chance that it may not occur.
(d)
The probability of an event is always a value between 0 and 1.

8.
A foreman in a manufacturing plant has three men and three women working for him. He wants to choose two workers for a special job. Not wishing to show any biases in his selection, he decides to select the two workers at random. What is the probability that at least one of the two selected workers will be a woman?
(a)
0.4
(b)
0.6
(c)
0.8
(d)
0.5
9.
Consider the following events in the roll of a single die:
A : Observe an odd number
B : Observe an even number
C : Observe a 1 or 2
D : Observe a 1
Which of the following is a pair of events that are independent?
(a)
A and B
(b)
A and C
(c)
B and D
(d)
A and D
10.
Diseases A and B are prevalent among people in a certain population. It is known that 20% of the population contracts disease A while 15% contracts disease B. Assuming these diseases are independent of each other, what percentage of the population contracts both diseases?
(a)
We cannot say.
(b)
10%
(c)
35%
(d)
3%
11.
There is a new game being played in Las Vegas; Roll 3 dice at the same time. If the total on the upfaces exceeds 11, you win $10. Otherwise, you lose $5. Before traveling to Las Vegas to play, you wish to estimate the probability of winning this game. You decide to perform 100 resampling trials. Which of the following resampling models would be the correct model to simulate the probability of winning.
(a)
Number of trials = 100
Minimum Value = 1
Maximum Value =18
Number to sample = 3
Without Replacement
(b)
Number of trials = 100
Minimum Value = 1
Maximum Value = 18
Number to sample = 3
With Replacement
(c)
Number of trials = 10
Minimum Value = 1
Maximum Value = 6
Number to sample = 3
With Replacement
(d)
Number of trials = 100
Minimum Value = 1
Maximum Value = 6
Number to sample = 3
With Replacement
12.
Roll 5 dice at the same time. If your total exceeds 18 you win. From the following results, estimate the probability of winning.
\epsfig{file=vegas518a.ps, width=5in, height=6in}

(a)
You will always lose.
(b)
0.8
(c)
You will always win.
(d)
0.2
13.
What is the error of your estimated probability?
(a)
. 253
(b)
.1265
(c)
.032
(d)
.8
14.
A baseball player needs 7 home runs or more in his last 20 at bats to win the home-run championship. The outcome of each at bat is either "home run" or "no home run". Historically, the probability that this player hits a home run during an at bat is .08. We also know that each at bat is independent of all the other at bats. Which of the following is the correct model to determine the probability that the player wins the championship? (i.e. which of the following inputs shown in the table below is correct)
\epsfig{file=baseballa.ps, width=6in, height= 5in}

(a)
Poisson Cumulative with k=6, lambda=20
(b)
Binomial Density with k=6, n=20, p=.08
(c)
Binomial Cumulative with k=6, n=20, p=.08
(d)
Poisson Density with k=6, lambda=20
15.
A psychologist is studying the stress level of air traffic controllers as determined by the arrival rate of the planes. The psychologist has determined that if there are 6 or more planes arriving within a 10 minute period, the air traffic controller will experience an unhealthy level of stress. At that particular airport, there are an average of 4 planes arriving in any 10 minute interval. Using the inputs and results shown below, which of the following is the correct calculation for the probability that the air traffic controller will not experience an unhealthy level of stress? This is what was input:
\epsfig{file=baseball2a.ps, width=6in, height= 5in}

       And these are the results for each of the submitted inputs:

     Rweb:> # CUMULATIVE BINOMIAL DISTRIBUTION  
     Rweb:> pbinom(4, 5, .8)  
     [1] 0.67232 

     Rweb:> # BINOMIAL PROBABILITY  
     Rweb:> dbinom(4, 5, .8)  
     [1] 0.4096 

     Rweb:> # CUMULATIVE POISSON DISTRIBUTION  
     Rweb:> ppois(5, 4)  
     [1] 0.7851304 

     Rweb:> # POISSON PROBABILIY  
     Rweb:> dpois(5, 4)  
     [1] 0.1562935
(a)
0.6723
(b)
0.2149
(c)
0.7851
(d)
0.1562
16.
Grade Point Averages for all WMU students follow a uniform probability distribution that range from 0.0 to 4.0. What type of random variable is Grade Point Average (G.P.A.)?
(a)
Uniform
(b)
Discrete
(c)
Continuous
(d)
Normal
The following is information for the next three questions. There was a fear that manatees are becoming extinct in Sutton Bay Florida. So a study was formed. One part of this study was a weekly flyover of the bay to obtain a count of the manatees observed. Below are the counts for 11 weeks:

 24  31  32  39  47  47  35  48  95  80  34
17.
Find the median of the manatee counts.
(a)
21.86
(b)
46.5.
(c)
39
(d)
11
18.
Find the quartiles of the manatee counts.
(a)
32, 48.
(b)
24, 95.
(c)
21.86, 46.5.
(d)
39, 46.5.
19.
Work through the outlier rule. What are the outliers in this data set?
(a)
24, 80 and 95.
(b)
95.
(c)
80 and 95.
(d)
No outliers.
The following is information for the next three questions. The sixth grade teacher Emma Austin is very disappointed in how her class did on the Meap tests last year. So this year she forced her students to read 5 classic novels. She had 60 students in each class so instead of listing these 120 test scores, she prepared the comparison boxplots shown below.

 
                           -----------
Last Year   ---------------I   +     I-------------      *
                           -----------
 
                               ------------
This Year   *    --------------I     +    I---------------
                               ------------
           +---------+---------+---------+---------+---------+------C10     
           0        20        40        60        80       100
20.
Determine the median scores for both years.
(a)
35, 52.
(b)
40, 60.
(c)
0, 94.
(d)
40, 52.
21.
Determine the lower adjacent point for This Year scores.
(a)
40.
(b)
12.
(c)
62.
(d)
94.
22.
Which of the following is the best conclusion for this data:
(a)
There are just too many outliers to make any statement concerning this data.
(b)
The students did seem to do better this year than last.
(c)
The students did about the same both years.
(d)
Students should be forced to read.
23.
Suppose the sample distribution of a large data set is right skewed with very large outliers on the right side. Which of the following statements would generally be correct?
(a)
The sample mean would lie to the right of the sample median.
(b)
The sample mean would lie to the left of the sample median.
(c)
The sample mean and the sample median would be practically the same.
(b)
There is just too much noise in the data to say anything.
24.
Suppose the registration office wants to compare the high school grade point averages of students at Central Michigan University to those at Western Michigan University. So we, as good statisticians, obtain random samples of size 100 from each university. We then analyze the samples and get ready to discuss our summary with the registration office. Which of the following plots would be most useful in our summary?
(a)
A boxplot of all the data (both samples combined)
(b)
Comparison boxplots of the samples.
(c)
A scatterplot of high school grade point averages versus height.
(d)
Individual boxplots, one for each sample.
25.
Bob takes a random sample on 100 students at Western Michigan University, asking them whether they are in favor or not of a policy for which each student must buy his/her own computer. He finds that 30% of his sample favors this policy. Which of the following is Bob's best conclusion?
(a)
Estimate that 30% of the students at WMU favor this policy but an estimate of the sampling error is also needed.
(b)
Estimate that 30% of the students at WMU favor this policy.
(c)
The sample size is too small to conclude anything.
(d)
There is just too much noise in the data to say anything.


 
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Stat 160
2002-02-18