Download Problem 4 (5 points)

Math 1070-2 Practice Final, December, 2004
Problem 1 (10 points)
Identify the only case in the following that does not constitute a distribution:
a.
b.
c.
Histogram
4.5
4
Frequency
3.5
3
2.5
Frequency
2
1.5
1
0.5
0
5
10
15
20
25
30
35
40
More
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d. X can only take two possible values and the probability of X=0 is 0.3 and the probability
of X=1 is 0.7.
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Problem 2 (5 points)
An instructor records the values of several variables for each student in her class. These include
the variables listed below. Which of these variables is categorical
a. Score on the final exam (out of 200 points).
b. Final grade for the course (A, B, C, D, or F).
c. The total number of points earned in the class (i.e., the total of the points on all exams
and quizzes in the course. The maximum number of points possible is 500).
d. The number of lectures the student missed.
Problem 3 (5 points)
When drawing a histogram it is important to
a. Have a separate class interval for each observation to get the most informative plot.
b. Make sure the heights of the bars exceed the widths of the class intervals so that the bars
are true rectangles.
c. Label the vertical axis so the reader can determine the counts or percent in each class
interval.
d. Make certain the mean and median are contained in the same class interval, so that the
correct type of skewness can be identified.
Problem 4 (5 points)
The histogram below gives the distribution of scores of 47 students in a test (of a possible total of
Histogram
8
7
Frequency
6
5
4
Frequency
3
2
1
or
e
0
0
0
0
0
0
0
M
16
15
14
13
12
11
10
90
80
70
60
0
50
170 points).
Which of the following is a correct statement?
a. Approximately half of the students scored between 90 and 120.
b. The histogram shows several strong outlier.
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c. The mean and the median are close to each other (within 10 points)
d. None of the above.
Problem 5 (15 points)
A sample was taken of the verbal GRE scores of 20 applicants to graduate school at a large
university. Below are the scores. For convenience, the data are ordered.
280
490
310
510
340
520
350
520
370
600
410
610
420
670
420
720
420
750
470
770
The mean applicant score is
a. 472.5
b. 480
c. 485
d. 497.5
The median applicant score is
a. 472.5
b. 480
c. 485
d. 497.5
The first quartile for the applicant is
a. 340
b. 390
c. 480
d. 605
The third quartile for the applicant is
a. 340
b. 480
c. 605
d. 640
If 25 points were added to each other, then the standard deviation of the new score would
a. be increased by 5.
b. Be increased by 25.
c. Be increased by 625.
d. Remain unchanged.
Problem 6 (5 points)
A normal density curve has which of the following properties?
a. It is symmetric.
b. It has a peak centered above its mean.
c. The spread of the curve is proportional to the standard deviation.
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d. All of the above.
Problem 7 (10 points)
The temperature at any random location in a kiln used in the manufacture of bricks is normally
distributed with a mean of 1000 and a standard deviation of 50F.
If bricks are fired at a temperature above 1125F, they will crack and must be disposed of. If the
bricks are placed randomly throughout the kiln, the proportion of bricks that crack during the
firing process is
a. 49.38%
b. 2.28%
c. 47.72%
d. 0.62%
Problem 8 (8 points)
The distribution of actual weights of 8 oz chocolate bars produced by a certain machine is
normal, with a mean of 8.1 oz and a standard deviation of 0.1 oz. What weight should be put on
the chocolate bar wrappers so that only 1% of bars are underweight?
a. 7.77 oz.
b. 8.33 oz.
c. 7.87 oz.
d. 8.23 oz.
Problem 9 (5 points)
Consider the scatterplot below
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1.8
1.6
1.4
Y
1.2
1
Series1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
X
The form of the relationship represented in the plot is best described as
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a.
b.
c.
d.
multi-linear.
Negative association.
Clusters.
All of the above.
Problem 10 (5 points)
The least-squares regression line is
a. The line that makes the square of the correlation in the data as large as possible.
b. The line that makes the sum of the squares of the vertical distances of the data points
from the line as small as possible.
c. The line that best splits the data in half, with half of the points above the line and half
below the line.
d. All of the above.
Problem 11 (8 points)
A researcher wishes to determine whether the rate of water flow (in liters per second) over an
experimental soil bed can be used to predict the amount of soil washed away (in kilograms). The
researcher measures the amount of soil washed away for various flow rates, and from these data
calculates the least-squares regression line to be
Amount of eroded soil = 0.4 + 1.3 x (flow rate)
The correlation between amount of eroded soil and flow rate would be
a. 1/1.3
b. 1.3
c. positive, but we cannot say what the exact value is
d. either positive or negative. It is impossible to say anything about the correlation from the
information given.
Problem 12 (10 points)
A business has two types of employees – managers and workers. Managers earn either $100,000
or $200,000 per year. Workers earn either $10,000 or $20,000 per year. The number of male and
female managers at each salary level and the number of male and female workers at each salary
level are given in the two tables below.
Managers
Male
Female
$100,00080
20
$200,00020
30
Workers
Male
$10,00030
$20,00020
The proportion of male managers who make $200,000 per year is
a. 0.067
b. 0.133
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Female
20
80
c. 0.200
d. 0.400
From these data we many conclude
a. the mean salary of female managers is greater than that of male managers.
b. The mean salary of males in this business is greater than the mean salary of females.
c. The mean salary of female workers is greater than that of male workers.
d. All of the above.
Problem 13 (10 points)
In a recent article it was reported that a random sample of children in grades 2-4, a significant
negative relationship was found between the amount of homework assigned and student
attitudes.
This is an example of
a. An experiment.
b. An observational study.
c. The establishing of a casual relationship through correlation.
d. A block design, with grades as blocks.
The amount of homework assigned is
a. An explanatory variable.
b. A responsive variable.
c. A confounding variable.
d. None of the above.
Problem 14 (5 points)
If the knowledge that an event A has occurred implies that a second event B cannot occur, the
events A and B are said to be
a. Independent
b. Disjoint
c. Collectively exhaustive
d. The sample space
Problem 15 (8 points)
The incomes in a certain large population of college teachers have a normal distribution with
mean $30,000 and standard deviation $6000. Nine teachers are selected at random from this
population to serve on a salary review committee. What is the probability that their average
salary exceeds $20,000?
a. 0.6915.
b. 0.4325.
c. 0.3085.
d. essentially 1.
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Problem 16 (8 points)
Suppose X is a random variable with the B(n  4, p 1/4) distribution. The probability that X is
less than or equal to 3 is
a. 0.9961.
b. 0.6836.

c. 0.3164.
d. 0.0039.
Problem 17 (10 points)
An agricultural researcher plants 400 pots with a new variety of corn. The average yield for these
plots is x  150 bushels per acre. Assume that the yield per acre for the new variety of corn
follows a normal distribution with unknown mean  and standard deviation   20 bushels per
acre.

A 95% confidence interval for  is


a. 150  1.00.
b. 150  1.96.
c. 150  1.92.
d. 150  19.6. 

 of the following would produce a confidence interval with a smaller margin of error than
Which
the95% confidence interval you computed above?
a. Plant only 100 plots rather than 400 because 100 are easier to manage and control.
b. Plant 500 plots rather than 100.
c. Compute a 90% confidence interval rather than a 95% confidence interval. The decrease
in confidence indicates that we have a narrower interval.
d. None of the above.
Problem 18 (8 points)
Suppose we want a 95% confidence interval for the average amount spent on entertainment
(movies, concerts, dates, etc.) by Freshman in their first semester at a large university. The
interval is to have a margin of error of $2, and the amount spent has a normal distribution with a
standard deviation   $30 . The number of observations required is closest to
a. 25.
b. 30.
c. 609.

d. 865.
Problem 19 (5 points)
We conduct a statistical test of hypotheses and find that the null hypothesis is statistically
 significant at level   0.05 . We may conclude


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That the test would also be significant at level   0.10.
That the test would also be significant at level   0.01.
Both a) and b) are true.
Neither a) nor b) is true.


Problem 20 (8 points)
a.
b.
c.
d.
The level of calcium in the blood of healthy young adults follows a normal distribution with
mean =10 milligrams per deciliter and standard deviation  =0.4. A clinic measures the blood
calcium of 25 healthy pregnant young women at their first visit for prenatal care. The mean of
these 25 measurements is x  9.6 . Is this evidence that the mean calcium level in the population
from which these women come is less than 10? To answer this, test the hypotheses


H0 :  10,Ha :  10.

The P-value of your test is
a. Less than 0.0002

b. 0.3085
c. 0.6170
d. greater than 0.99
Problem 21 (8 points)
An SRS of 80 laborers who use the services of a national temporary employment agency found
that in the past year the average number of days worked by these laborers was x  107 days, with
standard deviation s  45 days. Assume the distribution of the number of days worked in the
population of laborers using this employment agency is approximately normal, with mean  . Are
these data evidence that  has lowered from the value of 120 days of5 years ago? To determine
this, we test
 the hypotheses

H0 :  120,Ha :  120

a. What is the appropriate degrees of freedom for this test?
b. Based on the data, what is the value of the one-sample t statistic?
c. Give a range of the P-value for the one-sample t test, as accurately as possible?
d. Suppose the mean and standard deviation obtained were based on a sample of 100
laborers rather than 80. How would the P-value change?
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Problem 22 (8 points)
Do students tend to improve their Math SAT scores the second time they take the test? A random
sample of four students who took the test twice received the following scores:
student
First score
Second score
1
450
440
2
520
600
3
720
720
4
600
630
Assume that the change in MATH SAT score (second score – first score) for the population of
all students taking the test twice is normally distributed with mean  . A 95% confidence interval
for  is
a. 25.0  64.29.
b. 25.0  47.54.

c. 25.0  43.08.
d. 25.0  33.24.


Problem
23 (15 points)


A political
analyst was curious if younger adults were becoming more conservative. He decided

to see if the mean age of registered Republicans was lower than that of registered Democrats. He
selected an SRS of 100 registered Republicans from a list of registered Republicans and
determined the mean age to be x1=39 years, with a standard deviation s1=8 years. He also
selected an independent SRS of 100 registered Democrats from a list of registered Democrats
and determined the mean age to be x2 =40 years, with a standard deviation s2 =10 years. Let 1
and 2 represent the mean ages of the populations of all registered Republicans and Democrats,
 hypotheses
respectively. Supposethe researcher is interested in testing the
H0 : 1  2,Ha : 1 2




a. What is the numerical value of the standard error of the difference in sample means?
b. What is the numerical value of the t statistic?
c. What is the P-value for the test (use the conservative value for the degrees of freedom)?
Problem 24 (8 points)
A newspaper conducted a statewide survey concerning the 1998 race for governor. The
newspaper took a random sample (assumed it is an SRS) of 1200 registered voters and found that
640 would for the Republican candidate. Is this evidence that a clear majority of the population
would vote for the Republican candidate? To answer this, test the hypotheses
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H0 : p  0.5,Ha : p  0.5

The P-value of your test is
a. 0.4920
b. 0.0330
c. 0.0104
d. less than 0.0002.
Problem 25 (10 points)
Have attitudes toward investing in the stock market changed in recent years as the growth of
stocks has slowed? In 1995, a random sample of 100 adults that had investments in the stock
market found that only 20 said they were investing for the long haul rather than to become rich
or make quick profits. A random sample of 100 adults that had investments in the stock market
in 2002 found that 36 were investing for the long haul rather than to become rich or make quick
profits. Let p1 and p2 be the actual proportion of all adults with investments in the stock market
in 1995 and in 2002, respectively, that were investing for long haul.
A 99% confidence interval for p1 - p2 is
a. –0.157
  0.062
b. –0.157  0.122
c. –0.157  0.160
d. 0.157  0.062 

 evidence of an increase between 1995 and 2002 in the proportion of adults with
Is there

investments
in the stock market who are investing for the long haul? To determine this, you test

the hypotheses
H0 : p1  p2,Ha : p1  p2
The P-value of your test is
a. between 0.1 and 0.05

b. between 0.05 and 0.01
c. between 0.01 and 0.001
d. below 0.001.
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