Download Midterm Solutions

STAT 0200
SUMMER 2009
Midterm Solutions
This is a closed book-exam. You may use a calculator and one two-sided sheet of
notes. Mark all answers clearly and be sure to include work in cases where partial
credit may be awarded. Necessary tables can be found in your textbook.
Section 1: Multiple Choice: Circle the best response to each question.
1. A sample was taken of the salaries of 20 employees of a large company. The
following are the salaries (in thousands of dollars) for this year.
28 49 31 51 34 52 35 52 37 60 41 61 42 67 42 72 42 75 47 77
Suppose each employee in the company receives a $3,000 raise for next year (each
employee's salary is increased by $3,000). The standard deviation of the salary for
the employees will
a)
b)
c)
d)
be unchanged.
increase by 3000
be multiplied by 3000
increase by 3000
2. Let X denote the time taken for a computer connection to be established between
the terminal in an executive's office and the computer at a remote factory site. It is
known that X has a normal distribution with a mean of 15 seconds and a standard
deviation of 3 seconds. On 90% of the occasions the computer link is made in less
than
a)
b)
c)
d)
17.45
17.70
11.16
18.84
seconds
seconds
seconds
seconds
3. A consumer group surveyed the prices for a certain item in five different stores,
and reported the average price as $15. We visited four of the five stores, and found
the prices to be $10, $15, $15, and $25. Assuming that the consumer group is
correct, what is the price of the item at the store that we did not visit?
a)
b)
c)
d)
$10
$15
$20
$25
4. Which of the following is true for any density curve?
a)
b)
c)
d)
It is symmetric.
The total area under the curve is 1.
It must either steadily rise or steadily fall. It cannot do both.
Bars must be of equal width.
5. A set of data has a median that is much larger than the mean. Which of the
following statements is most consistent with this information?
a)
b)
c)
d)
A histogram of the data is symmetric.
A histogram of the data is skewed left.
A histogram of the data is skewed right.
The data set must be so large that it would be better to draw a boxplot than
a histogram.
6. When creating a scatterplot, one should
a)
b)
c)
d)
use the horizontal axis for the response variable.
use the horizontal axis for the explanatory variable.
use a different plotting symbol if the explanatory variable is categorical than if
the response variable is categorical.
use a plotting scale that makes the overall trend roughly linear. .
7. Which of the following is true of the least-squares regression line?
a)
b)
c)
d)
The slope is the change in the response variable that would be predicted by a
one-unit increase in the explanatory variable.
It always passes through the point ( x , y ), where x and y are the means of
the explanatory and response variables, respectively.
It will only pass through all the data points if r = 1 or -1.
All of the above.
8. Suppose a straight line is fit to data having response variable y and explanatory
variable x. Predicting values of y for values of x within the range of the observed
data is called
a)
b)
c)
d)
contingency.
interpolation.
causation.
correlation.
9. A researcher observes that, on average, the number of divorces in cities with
major league baseball teams is larger than in cities without major league baseball
teams. The most plausible explanation for this observed association is
a)
b)
c)
d)
the presence of a major league baseball team causes the number of divorces
to rise (perhaps husbands are spending too much time at the ballpark).
the high number of divorces is responsible for the presence of a major league
baseball team (more single men means potentially more fans at the ballpark,
making it attractive for an owner to relocate to such cities).
the association is due to common response (major league teams tend to be in
large cities with more people hence a greater number of divorces).
the observed association is purely coincidental. It is implausible to believe the
observed association could be anything other than accidental.
10. Exploratory data analysis refers to
a)
b)
c)
d)
the use of graphs, tables, and numbers to uncover patterns and relationships
in data.
techniques for collecting data to answer specific questions.
techniques for answering specific questions from data with a known degree
of confidence.
all of the above.
11. A television station is interested in predicting whether voters are in favor of an
increase in the state sales tax. It asks its viewers to phone in and indicate whether
they support or are opposed to an increase in the state sales tax in order to
generate additional revenue for education. Of the 2633 viewers who phoned in,
1474 (55.98%) were opposed to the increase. The number 55.98% is
a)
b)
c)
d)
a
a
a
a
statistic.
parameter.
sample.
population
12. A random variable is
a)
b)
c)
d)
a numerical outcome of a random phenomenon.
any numerical variable with the property that the probability of any particular
fixed outcome varies.
a numerical variable whose sample space is totally random.
a variable whose outcomes have probabilities that are both numerical and
random.
13. Event A occurs with probability 0.5. The probability that disjoint events A or B
occurs 0.8. The probability that event B occur is
a)
b)
c)
d)
0.3.
0.4.
0.8.
cannot be determined from the information given.
14. A study found a correlation of r = -0.61 between the gender of a worker and his
or her income. You may correctly conclude
a)
b)
c)
d)
women earn more than men on the average.
women earn less than men on the average.
an arithmetic mistake was made. Correlation must be positive.
this is incorrect because r makes no sense here.
15. I select two cards from a deck of 52 cards and observe the color of each (26
cards in the deck are red and 26 are black). Which of the following is an appropriate
sample space S for the possible outcomes?
a)
b)
c)
d)
S = {red, black}
S = {(red, red), (red, black), (black, red), (black, black)}, where, for
example, (red, red) stands for the event "the first card is red and the second
card is red."
S = {1, 2}
All of the above.
16. Suppose that A and B are disjoint events with P(A)=0.2, P(B)=0.4.
Then P(A or B) is
a)
b)
c)
d)
0.10.
0.20.
0.30.
0.60.
17. Consider two events A and B. The probability of event A not occurring is .6.
The probability of event B not happening is .7. Events A and B are disjoint. What is
the probability that either events A or B occur?
a)
b)
c)
d)
.35
.50
.65
.70
Section 2: Workout Problems: Be sure to show all work in order to receive
partial credit.
18. Farmers are concerned about the length of the growing season in their part of
the country. An agronomist measures the length of the growing season (number of
days between the last frost in the spring and the first frost in the fall) in a sample of
regions. She used MINITAB to calculate descriptive statistics for the data. Here is
the output:
GROW
N
57
MEAN
203.65
MEDIAN
192.00
TRMEAN
200.92
GROW
MIN
125
MAX
335
Q1
167.50
Q3
233.00
STDEV
51.87
SEMEAN
6.87
Use the output to answer the following questions:
a.
How many regions were in the sample? _________57__________
b.
25% of regions have growing regions of less than _______167.50_____ days.
c.
What is the variance of this data set (give the numerical value)?
d.
Is this data set reasonably symmetric, positively skewed, or negatively
skewed?
positively skewed because
e.
Construct a boxplot of the data below. (Be sure to include the scale of
numbers on an axis.)
125
167.5 192
233
335
19. Suppose Z is a random variable which follows a standard normal distribution.
a.
What proportion of observations of Z are greater than -0.13?
b.
What proportion of observations of Z are between -0.29 and 1.45?
c.
What is the twentieth percentile of Z? [In other words, find the value of z
such that approximately 20% of standard normal observations fall below it.]
d.
What is the third quartile of z? [In other words, find the value of z such that
the proportion of standard normal observations falling below z is
approximately 75%.]
20. The weights of newborn children in the United States vary according to a normal
distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The
government classifies a newborn as having low birth weight if the weight is less than
5.5 pounds.
a.
What proportion of babies weigh less than 5.5 pounds at birth?
b.
If the government wanted to change the value 5.5 pounds to a weight where
only 2% of newborns weigh less than the new value, what weight should
they use?
21. The owner of a small video rental store claims that the average number of
videotapes rented daily (y) is related to the rental price charged per tape per day
(x). She collected data for six different prices. The data and a scatterplot are
presented below.
PRICE:
RENTALS:
3
227
2.75
240
2.5
270
2.25
290
2
320
1.75
380
400
350
rentals
300
250
200
150
100
50
0
0
1
2
3
4
price
Summary statistics from the data are:
x = 2.375
y = 287.8333
sx = 0.4677
sy = 56.2865
r =-.97338
a.
What is the equation of the least-squares regression line?
b.
If the owner charges $2.20 to rent a video, what is the predicted number of
videos she will rent? Do not round your answer to the nearest integer.
c.
What is the value of the residual for the data value ($2.50, 270)?
d.
Describe the strength, form, and direction of the association between the
rental price of a video and the number of videos rented per day. Can we
conclude that an increase in the rental price causes the number of videos
rented per day to decrease?
Strong, negative and linear but slightly curved association.
22. In a shipment of 15 room air conditioners, there are 4 with defective
thermostats. Two air conditioners will be selected at random (without replacement)
and inspected one after the other. Find the probability that:
a.
The first is defective.
b.
The first is defective and the second one is good.
c.
Both are defective.