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
**(b).****(c).****(d).**

- 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
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 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)?

**(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
.
**(a).**-
*Pr*(*s*_{1}) = 0.5,*Pr*(*s*_{2}) = 0.7,*Pr*(*s*_{3}) = -0.2 **(b).**-
*Pr*(*s*_{1}) = 0.4,*Pr*(*s*_{2}) = 0.4,*Pr*(*s*_{3}) = 0.4 **(c).**- X
*Pr*(*s*_{1}) = 0.25,*Pr*(*s*_{2}) = 0.5,*Pr*(*s*_{3}) = 0.25 **(d).**-
*Pr*(*s*_{1}) = 2,*Pr*(*s*_{2}) = 1,*Pr*(*s*_{3}) = 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 DistributionPayout (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

- 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

- 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,
*R*^{2}?**(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
(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