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Stat 160 Midterm Exam
Form A

$\;$

Feb. 26, 2002

Attempt all problems.

1.
At a service window at the bank, there can be at most 3 people waiting. Suppose the probability that no one is waiting is .2; that one person is waiting is .3; that 2 people are waiting is .4; and that 3 people are waiting is .1. Determine the probability that at least 2 people are waiting.
(a)
.4
(b)
.9
(c)
.5 X
(d)
.6
2.
In the last problem compute the mean ($\mu$) of the number people waiting.
(a)
1.0
(b)
1.5
(c)
1.4 X
(d)
.4
3.
Suppose John is a professional basketball player who makes 65% of his free throws. Suppose we are interested in the probability that he makes 12 of his next 15 free throws.

Figure 1 below gives the class code for a binomial problem. Note that it requires whether the choice of cumulative or density (exact); the number of successes k; the number of trials n; and the probability of success p.


  
Figure 1: Class Code Page for Binomial and Normal Probabilities
\begin{figure}
\centering
\leavevmode
\epsfig{file = probcmds.ps, height=5in, width = 5in, angle=0} \end{figure}

Choose the correct values of these items to determine the above probability that John makes 12 of his next 15 free throws.

(a)
Binomial density; k=15; n=12; p=.65
(b)
Binomial cumulative; k=12; n=15; p=.65
(c)
Binomial density; k=15; n=12; p=.65
(d)
Binomial density; k=12; n=15; p=.65 X

4.
For the situation in the last problem, suppose we are interested in the probability that he makes at least 9 of his next 20 free throws. Select the right set of values which will solve this problem.
(a)
Binomial cumulative; k=8; n=20; p=.65 X
(b)
Binomial density; k=9; n=20; p=.65
(c)
Binomial cumulative; k=9; n=20; p=.65
(d)
Binomial density; k=8; n=20; p=.65

5.
For the situation in the last problem, determine the mean and variance of the number of successful free throws out of 20.

(a)
$\mu = 13$, $\sigma^2 = 4.55$. X
(b)
$\mu = 20$, $\sigma^2 = .65$.
(c)
$\mu = .65$, $\sigma^2 = 20$.
(d)
$\mu = 13$, $\sigma^2 = 2.13$.

6.
The scores on the entrance exam to a prestigious college are normally distributed with mean 75 and standard deviation 10. Suppose we are interested in the probability that a student scores at least 80 on the exam (the score needed to apply to the college). The class code for questions on normal probabilities is given in the Figure 1. Notice that it requires a choice on cumulative normal or percentage; selection of k; selection of mu ($\mu$); and selection of the Std. Dev. ($\sigma$). Determine the correct values which will lead to the probability that a student scores at least 80 on the exam.

(a)
Cumulative Normal; k=75; $\mbox{mu} =80$; $\mbox{Std.\ Dev.\ } = 10$.
(b)
Normal Percentage; p=80; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$.
(c)
Cumulative Normal; k=80; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$. X
(d)
Normal Percentage; p=75; $\mbox{mu} =80$; $\mbox{Std.\ Dev.\ } = 10$.

7.
For the situation in the last problem, suppose that we want to find the 90th percentile of the scores on the exam; i.e., the score that only the top 10% of the students exceed. Determine the correct values which will lead to this score.

(a)
Normal Percentage; p=.90; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$. X
(b)
Cumulative Normal; k=.90; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$.
(c)
Normal Percentage; p=.10; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$.
(d)
Cumulative Normal; k=.10; $\mbox{mu} =75$; $\mbox{Std.\ Dev.\ } = 10$.
8.
Suppose the height of an adult female is normally distributed with mean 64 inches and standard deviation 4 inches. Use the Class Code output below to determine the probability that a woman is between 62 and 69 inches tall.

  Rweb:> # CUMULATIVE NORMAL DISTRIBUTION
  Rweb:> pnorm(64, 69, 4)
  [1] 0.1056498

  Rweb:> # CUMULATIVE NORMAL DISTRIBUTION
  Rweb:> pnorm(62, 64, 4)
  [1] 0.3085375

  Rweb:> pnorm(62, 69, 4)
  [1] 0.04005916

  Rweb:> # CUMULATIVE NORMAL DISTRIBUTION
  Rweb:> pnorm(69, 64, 4)
  [1] 0.8943502
(a)
.5858 X
(b)
.8543
(c)
.6800
(d)
.9500

9.
You have a drawer with three colors of unsorted socks: black, white, and purple. There are 6 black socks, 8 white socks and 2 purple socks. On a dark morning, you reach into your drawer and randomly pull out two socks. What is the probability you have a matched pair of socks. (black-black, white-white, or purple-purple) HINT: Use a tree diagram.

(a)
.33
(b)
.37 X
(c)
.67
(d)
.023
10.
Which of the following statements is false:
(a)
The probability of the sample space is always 1.
(b)
A negative probability is possible, but only when sampling without replacement. X
(c)
An Event's probability and the probability of an Event's complement always sum to 1.
(d)
It is possible to have a probability of 0.
11.
According to a national survey, the probability of a husband and wife watching TV together is 25%. The survey also determined that the probability of the husband watching TV is 60% and the probability of the wife watching TV is 50%. Let Event A = {the husband is watching TV} and let Event B = {the wife is watching TV}. Based upon the results of this survey, are Event A and Event B independent of each other?
(a)
Cannot tell from the information given
(b)
Yes
(c)
Only if the husband and wife are home together at 9 pm.
(d)
No X

12.
When performing resampling trials, what happens to our probability estimate as the number of trials increases (say from 10 to 100).
(a)
The error of our estimate increases.
(b)
Error stays the same.
(c)
Both A and B are true
(d)
The error of our estimate decreases. X
13.
A new type of missile is to be tested. It has been estimated that the probability of a successful missile launch is 80%. Each launch has been prepared by a different team of technicians, so each launch can be considered independent of the other. If four launches are made, what is the probability of zero (0) successful launches?
(a)
.9984
(b)
.0016 X
(c)
.4096
(d)
.5904

14.
A softball player needs 2 hits in her last 3 at bats to win the batting championship. At each at bat, there is a .4 probability of getting a hit. (she's hitting .400 for the season) and each at bat can be considered independent of the previous one. What would be the correct RWEB input to perform 1000 resampling trials in order to estimate the probability of her winning the batting championship. Let the numbers 0-4 be considered a hit and 5-9 be considered no hit.
(a)
ID number: 123 Number of Trials: 3 Minimum Value: 0 Maximum Value: 1000 Number to be drawn: 9 Without Replacement
(b)
ID number: 123 Number of Trials: 9 Minimum Value: 0 Maximum Value: 1000 Number to be drawn : 3 Without Replacement
(c)
ID number: 123 Number of Trials: 1000 Minimum Value: 0 Maximum Value: 4 Number to be drawn: 1000 With Replacement
(d)
ID number 123 Number of Trials: 1000 Minimum Value: 0 Maximum Value: 9 Number to be drawn: 3 With Replacement X
15.
You are a United States Senator from Michigan, a state with 5 million voters. A random sample of Michigan citizens surveyed by your staff has estimated that 70% of your voters approve of increased military funding. An important vote regarding military funding is coming up and you always vote with the majority of your voters. What decision will you make regarding your vote?
(a)
Vote "YES" for increased funding. Clearly 70% is a majority and the error will make no difference.
(b)
We need to know the error of this estimate before any final decisions are made. X
(c)
Vote "NO" against increased funding. 70% in favor of funding will not be a majority after accounting for the error.
(d)
Survey all 5 million of your voters.
16.
You roll a fair 6 sided die one time. Let Event A = {you roll a 6}. What is the complement of Event A?
(a)
Roll a 6
(b)
Roll a 1,2,3,4, or 5 X
(c)
Roll a 1
(d)
Event B
The following two questions refer to the following stem and leaf plot of midterm test scores obtained from 19 students of a course in elementary statistics
The decimal point is 1 digit(s) to the right of the |
      4 | 8
      5 | 2578
      6 | 01245689
      7 | 0235
      8 | 2
17.
What is the shape of the data set?
(a)
Right Skewed
(b)
Bimodal
(c)
Left skewed
(d)
Symmetric X
18.
What is the range of the midterm scores?
(a)
1.2
(b)
34 X
(c)
12
(d)
64.5
The following four questions refer to the following situation. The marketing manager of a large supermarket chain would like to determine the effect of shelf space (in feet) on the sales (in thousands of dollars) of pet food, using the least squares method. A random sample of 12 equal-sized stores is selected. From this random sample, the following scatter diagram and least squares fit were obtained.

\epsfig{file = scatter1.ps, height=5in, width = 5in, angle=-90}

                         Parameter       Standard
     Variable            Estimate         Error    t Value    Pr > |t|
     Intercept            1.45000        0.21783     6.66      <.0001
     Shelf Space          0.07400        0.01591     4.65      0.0009
19.
Which linear model appropriately answers the objective?
(a)
Sales = 1.45 + 0.074*Shelf Space X
(b)
Sales = 0.074+ 1.45*Shelf Space
(c)
Shelf Space = 0.074 + 1.45*Sales
(d)
Shelf Space = 1.45 + 0.074*Sales
20.
What does the estimate of the slope mean in terms of this problem?
(a)
We expect sales to increase by 1.45 thousand dollars for every foot increase in shelf space.
(b)
We expect shelf space to increase by 0.074 foot for every thousand dollar increase in sales.
(c)
We expect sales to increase by 0.074 thousand dollars for every foot increase in shelf space. X
(d)
We expect shelf space to increase by 1.45 feet for every thousand dollar increase in sales.
21.
Predict the weekly sales (in hundreds of dollars) of pet food stores with 8 feet of shelf space for pet food. Is this a valid prediction?
(a)
1.67, a valid prediction
(b)
2.04, a valid prediction X
(c)
2.04, an invalid prediction
(d)
1.67, an invalid prediction
22.
Which of the following is the most probable value of the correlation coefficient between Sales and Shelf Space? How much variation in Sales can be explained by Shelf Space?
(a)
0.83, 69% of the variation in Sales can be explained by Shelf Space. X
(b)
0, 50% of the variation in Sales can be explained by Shelf Space
(c)
-0.83, 83% of the variation in Sales can be explained by Shelf Space
(d)
1, 100% of the variation in Sales can be explained by Shelf Space
23.
Crazy Dave, a well-known baseball analyst, would like to study the various team statistics for the 1997 baseball season to determine which variables might be useful in predicting the number of wins achieved by teams during the season. He has decided to begin by using the team earned run average (ERA) to predict the number of wins using a Wilcoxon fit. The following is the residual plot of the fit.

\epsfig{file = resid1.ps, height=5in, width = 5in, angle=-90}

Is the Wilcoxon fit a good fit for the data?

(a)
No, since the residuals are not randomly scattered around the zero line. X
(b)
Yes, since the residuals are randomly scattered around the zero line.
(c)
Yes, since the variance of the residuals is increasing.
(d)
B and C are true.
The next two questions refer to the following situation. The operations manager of a plant that manufactures tires wishes to obtain the average inner diameter of a grade of tire. A random sample of 6 tires was selected, and the results representing the inner diameters of the tires were as follows: 568, 570, 575, 578, 580, 600.

24.
Are there any outliers in the data set?
(a)
No outlier is present in the data set.
(b)
Only 600 is an outlier. X
(c)
568 and 600 are outliers.
(d)
Only 568 is an outlier.
25.
The manager decided to use the Hodges-Lehmann estimator to estimate the average (center) of the data. What is its value?
(a)
577.5
(b)
575
(c)
574.5
(d)
576.5 X



 
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Stat 160
2002-02-28