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Form A MidTerm Stat 160 Fall Term 2002 Oct. 24, 2002 NAME:

1.
Roll 3 dice at the same time. If your total exceeds 12 you win. From the following results, estimate the probability of winning.

   Number of trials = 10
   Minimum value = 1
   Maximum value = 6
   Number to sample = 3
   With Replacement

Trial 1
4       6       6

Trial 2
3       3       2

Trial 3
3       4       6

Trial 4
1       3       1

Trial 5
1       2       5

Trial 6
2       2       3

Trial 7
5       1       3

Trial 8
1       4       6

Trial 9
5       3       1

Trial 10
5       4       6

(a).
X $\widehat{p}= 0.30 $
(b).
$\widehat{p}= 0.70 $
(c).
$\widehat{p}= 0.21 $
(d).
$\widehat{p}= 0.10 $

2.
What is the error of the estimated probability for the previous problem?.

(a).
X 0.289
(b).
0.145
(c).
0.042
(d).
0.458

3.
The number of automobiles exiting a busy interstate highway between 5-5:15 pm follows a Poisson distribution with a mean of $\lambda=500$ automobiles. This exit ramp has a capacity of 600 autos, otherwise a line will form on the highway. Let X be a random variable representing the number of autos exiting the exit ramp. What is the probability of a line forming on this exit ramp between 5-5:15 pm (in other words, what is the probability X > 600)? Use the RWEB output shown below.

Rweb:> # CUMULATIVE POISSON DISTRIBUTION
 Rweb:> ppois(600, 500)
[1] 0.9999936

 Rweb:> # POISSON PROBABILITY
 Rweb:> dpois(600, 500)
[1] 1.356671e-06

Rweb:> # CUMULATIVE POISSON DISTRIBUTION
 Rweb:> ppois(500, 600)
[1] 1.484134e-05

(a.)
.9999936
(b.)
.000001356671
(c.)
.0000148134
(d.)
X .0000064





The next two questions pertain to the following situation:

The United States Army has developed a new missile that will hit its target 90% of the time. During a battle, 200 of these new missiles were launched. Using the RWEB output shown below and your knowledge of the binomial distribution, please answer the following questions:

RWEB Output:

Rweb:> # CUMULATIVE BINOMIAL DISTRIBUTION
 Rweb:> pbinom(200,190,.9)
[1] 1

Rweb:> # BINOMIAL PROBABILITY
 Rweb:> dbinom(200, 190, .9)
[1] 0

Rweb:> # CUMULATIVE BINOMIAL DISTRIBUTION
 Rweb:> pbinom(190,200,.9)
 [1] 0.9964714

Rweb:> # BINOMIAL PROBABILITY
 Rweb:> dbinom(190, 200, .9)
 [1]0.004542684

4.
What is the probability that more than 190 missiles will hit their target?
(a.)
.004542684
(b.)
X .0035286
(c.)
1
(d.)
.9964714

5.
What is the expected number of missiles that will hit their target?

(a.)
90
(b.)
X 180
(c.)
190
(d.)
200

6.
A class consists of 30 students, of whom 10 are men and 20 are women. Five of the women and none of the men are out-of-state students. A student is selected at random from class. What is the probability that the one selected will be an out-of-state student?.

(a).
0.667
(b).
0.333
(c).
X 0.17
(d).
0.5

7.
Data were obtained on the peak power load for a power plant. The 5 basic Descriptive Statistics for the data are given below
 Min=129 Q1=153 Md=207 Q3=213.8 Max=266

Compute the upper fence for these data and determine whether or not the maximum is an outlier.

(a)
305.0, is an outlier
(b)
X 305.0, not an outlier
(c)
91.2, not an outlier
(d)
91.2, is an outlier

8.
Consider the statements:

I. If $Pr(E)*Pr(F) = Pr(E \& F) = P(E \cap F)$ then the sets E and F are independent.

II. If Pr(E|F) = Pr(E) and Pr(F|E) = Pr(F) then the sets are independent

(a).
I and II are both false
(b).
Only II is true
(c).
X I and II are both true
(d).
Only I is true

9.
You are playing darts at a local pub. The dart board has 3 zones (1-bullseye,2-middle ring,3-outer ring) as shown in the picture below. Zone 1 has an area of 1 square inch, Zone 2 has an area of 3 square inches, and Zone 3 has an area of 6 square inches. Say a dart is thrown at random and it hits the dart board. What is the probability of hitting Zone 1 (bullseye)?
\begin{figure}
\centering
\epsfig{file=dart.EPS,width=5in,height=4in} \end{figure}

(a.)
1 out of 6 (.167)
(b.)
1.00
(c.)
X .10
(d.)
None of the above.

10.
Suppose we have data on student's weekly allowance which is right-skewed with a very large outliers on the right side. Which of the following statements will be generally correct?
(a)
The sample mean weekly allowance and sample median weekly allowance will be the same.
(b)
The sample mean weekly allowance would lie to the left of the sample median weekly allowance.
(c)
X The sample mean weekly allowance would lie to the right of the sample median weekly allowance.
(d)
The sample median weekly allowance would lie to the right of the sample mean weekly allowance.

11.
Which of the following probabilities are feasible for an experiment having sample space $\{s_1, s_2, s_3\}$.

(a).
Pr(s1) = 0.5, Pr(s2) = 0.7, Pr(s3) = -0.2
(b).
Pr(s1) = 0.4, Pr(s2) = 0.4, Pr(s3) = 0.4
(c).
X Pr(s1) = 0.25, Pr(s2) = 0.5, Pr(s3) = 0.25
(d).
Pr(s1) = 2, Pr(s2) = 1, Pr(s3) = 0.5





The next three questions pertain to the following situation:

The yearly harvest of the Michigan apple crop (in millions of bushels) follows a normal distribution with a mean of 10 and a standard deviation of 3. Using the RWEB output shown below, determine the following probabilities:

RWEB Output:

Rweb:> # CUMULATIVE NORMAL DISTRIBUTION
Rweb:>pnorm(.9, 10, 3)
 [1] 0.001209341

Rweb:> # NORMAL PERCENTAGE POINT
Rweb:>qnorm(.9,10, 3)
 [1] 13.84465

 Rweb:> # CUMULATIVE NORMAL DISTRIBUTION
Rweb:> pnorm(16, 10, 3)
 [1] 0.9772499

Rweb:> # NORMAL PERCENTAGE POINT
 Rweb:> qnorm(.16, 10, 3)

 [1] 7.016626

 Rweb:> #CUMULATIVE NORMAL DISTRIBUTION
 Rweb:> pnorm (10,10,3)
 [1] .500

12.
90% of the time, the apple crop will be less than what amount (in millions of bushels)?
(a.)
.9772
(b.)
7.01
(c.)
X 13.84
(d.)
.001209

13.
What is the probability that the Michigan apple crop harvest is more than 16 (million) bushels?
(a.)
13.84
(b.)
.001209
(c.)
.9772
(d.)
X .0227

14.
What is the probability that the Michigan apple crop will be between 10 and 16 (million) bushels?
(a.)
.0012
(b.)
.500
(c.)
.9772
(d.)
X .4772

15.
A new hand calculator is designed to be ultrareliable by having two independent calculating units. The probability that a given calculating unit fails within first 1000 hours of operation is .001. What is the probability that at least one calculating unit will operate without failure for the first 1000 hours of operation?.

(a).
0.000001
(b).
X 0.999999
(c).
0.001
(d).
0.998

16.
The Michigan Lottery's new $1 instant game has a payout distribution given in the table shown below. Note that a minus sign indicates money given to a winner, while a plus sign indicates money kept by the Michigan Lottery. Let X be the payout (in dollars). Given the distribution of X as shown in the table, what is the expected value of X (the amount of money the Michigan Lottery will make (or lose) on each ticket sold)?


 
Table 0.1: Lottery Payout Distribution
Payout (X) +$1 -$2 -$5 -$10 -$30
Probability (X) .9 .04 .03 .02 .01

(a.)
Lose $1.63
(b.)
Lose $48
(c.)
Make $1
(d.)
X Make $ 0.17





The next two problems are concerned with the following situation:

Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who desire a low-impact aerobics program. In a study to investigate the relationship between exercise cadence and heart rate, the heart rates of 20 healthy volunteers were measured at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are summarized in the stem-and-leaf plot


\begin{verbatim}7\vert 89
8\vert17
9\vert1244566688
10\vert 179
11\vert 2
\par\end{verbatim}

17.
What heart rates are the first and third quartiles?
(a)
X 87 and 98
(b)
80 and 90
(c)
78 and 112
(d)
93.7 and 94.5

18.
What is the median heart rate of the 20 volunteers?
(a)
94.0
(b)
X 94.5
(c)
95.0
(d)
93.7





The next three problems are concerned with the following situation:

The following table gives the data on mathematics achievement test scores and calculus grades for ten independently selected college freshmen.

Math Achievement Test Score Final Calculus Grade
39 65
43 78
21 52
64 82
57 92
47 89
28 73
75 98
34 56
52 75

Suppose we are interested in predicting the final calculus grade of a student in terms of his/her math achievement test score. The least squares procedure gave

\begin{displaymath}\mbox{$\widehat{Y} = 40.7842 +0.7656X $\space and $R^{2}=70.52$ }\;.
\end{displaymath}

19.
Using the regression equation (if possible), predict the final calculus grade given a math achievement test score of 50.
(a)
40.7842
(b)
12.0374
(c)
X 79.0642
(d)
98.0000

20.
Which of the following is the best interpretation of the slope of the regression equation?
(a)
When the math score is equal to zero, final calculus grade is 40.7842.
(b)
When the math score increases by 0.7656, final calculus grade increases by 40.7842.
(c)
It has no practical meaning as x=0 is not in the range of the data.
(d)
X When the math score increases by 1, final calculus grade increases by 0.7656.

21.
Which of the following is the best interpretation of the coefficient of determination, R2?
(a)
When the math score is equal to zero, final calculus grade is 70.52
(b)
For every 1 point increase in math score, final calculus grade increases by 70.52.
(c)
X 70.52% of variation in final calculus grade is accounted for by math achievement test score.
(d)
29.48% of variation in final calculus grade is accounted for by math achievement test score.

22.
A random sample of incoming WMU freshmen indicates that their high school Grade Point Average (GPA) is normally distributed with a mean of 2.75 and a standard deviation of .25. Using the empirical rule, what is the range of the middle $\frac{2}{3}$ (67%) of all incoming WMU freshmen?

(a.)
1.00-4.00
(b.)
2.25-3.25
(c.)
X 2.50-3.00
(d.)
0.25-2.75

23.
Suppose 20 passengers are on a bus that enters a foreign country. 12 of these passengers are women. At the gate to the foreign country, a guard gets on the bus and selects 6 people at random for an extensive visa check. We want to estimate the probability that all 6 are males. You decide to perform 100 resampling trials. Which of the resampling models would be correct to estimate the probability?

(a).
X
   Number of trials = 100
   Minimum value = 1
   Maximum value = 20
   Number to sample = 6
   Without Replacement

(b).
   Number of trials = 10
   Minimum value = 1
   Maximum value = 20
   Number to sample = 6
   Without Replacement

(c).
   Number of trials = 100
   Minimum value = 0
   Maximum value = 20
   Number to sample = 1
   Without Replacement

(d).
   Number of trials = 100
   Minimum value = 1
   Maximum value = 20
   Number to sample = 6
   With Replacement

24.
The following are the weights of a small sample of students. Obtain the Hodges-Lehmann estimate of location for these data.
    150   180   200   260

(a)
46.458
(b)
197.5
(c)
190
(d)
X 195

25.
From an ordinary deck of playing cards, four cards are to be drawn successively at random and without replacement. Find the probability of receiving in order a spade, a heart, a diamond, and a club.

(a).
X 0.0044
(b).
0.0039
(c).
0.0023
(d).
0.0026



 
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Stat 160
2002-10-29