Review Sheet for the Final Exam in MATH 108
Fall 2012
A. This sheet is only a review of some of the most important material from the course, but not
necessarily a review of all of the course material. It is a great starting point from which you can study for
the final exam, but not necessarily the only document from which you should study for the final exam.
B. Unless it is indicated on a problem specifically, you have an option of using one of the two
methods on any of the problems below: a) “by hand” computations (including the algebraic buttons of your
calculator, such as , , etc.), or b) statistical commands of your graphing calculator.
C. Always show your work: a) if a problem is done by hand, write appropriate formulas,
substitutions of numbers into formulas, then compute the answer; b) if a problem is done using statistical
features of your calculator, write down which commands/buttons you used and what values you entered, so
that partial credit may be awarded, especially in case of an incorrect final answer.
D. The exact answers are the most accurate. However, values rounded to 4 decimal places on
probability answers and values on the horizontal axis (for instance, z-scores) rounded to 2 decimal places
will also be accepted.
E. Problems marked with a * symbol denote the most important problems for the course.
1. A random sample of 8 high school students was selected. Each student was asked how
much time he or she spent watching TV during the previous week. The following times
(in hours) were obtained: 4.3, 5.0, 5.5, 6.0, 7.5, 10.0, 11.0, 11.5
a) Construct a relative frequency distribution (table) for these data, grouping them into
classes with class width of 2 with the first class having the lower limit of 4. Use the
convention of including the left endpoint, but not the right endpoint of each class.
b) Construct a relative frequency histogram for these data using your answer to part a).
c) Identify whether the variable X=”the time spent watching TV last week” is qualitative,
quantitative discrete, or quantitative continuous.
2. Given the sample of 3, 0, 7, -2, without using the statistical features of your calculator,
compute its a) sample mean, b) sample variance, c) sample standard deviation.
3. Given the population of 5, 10, 15, 20, 25, compute its a) mean, b) standard deviation.
4. a) If X is the variable that denotes the price of a home sold in NJ in 2006, then its mean
is (circle one): i) higher than the median, ii) lower than the median, iii) equal the median.
b) If X=“the annual salary of a National Football League player in 2008” then the
following is a better measure of center for X (circle one): i) the mean, ii) the median.
5. In her first semester of college, Rebecca has completed a 4 credit English course and
got a D, a 2 credit art course and got an A, a 4 credit Math course and got a B, and a 3
credit sociology course and got a C. What is Rebecca’s grade point average for that
semester? Assume A counts as 4 points, B for 3 points, etc.
6. The mean height of a man is 69.6 in. with standard deviation of 3.0 in. and the mean
height of a woman is 64.1 in. with standard deviation of 3.8 in. Who is relatively taller in
their gender group, a 74 inch man or a 72 inch woman? Justify your answer.
7. The following measurements represent heights, in inches of randomly selected men:
68, 69, 70, 70, 71, 72, 64, 65, 65, 67, 68, 73, 73, 75, 76.
a) Find and interpret the 60th percentile of this sample.
b) Find and interpret the 1st quartile of this sample.
c) Find the sample median.
8. The following data represent a study of the relationship between the number of hours a
student studies per week for a math course and the numerical course grade he or she
received.
# of study hours 2 4 5.5 6 6.5 9
Grade
50 62 80 75 90 95
a) Which one, the number of study hours, or the grade, should be used as the independent
(that is, the explanatory) variable and which one should be used as the dependent (that is,
response) variable?
b) Draw a scatter diagram for these data based on your decision in a). Does the
relationship between the two variables appear linear?
c) If your answer to b) is yes, compute the linear correlation coefficient for these data.
Comment on your answer.
d) Compute the equation of the linear regression line and graph the line on your scatter
diagram. Make sure to label the values of at least two points on the line.
e) Predict the numerical grade of someone who studies 5 hours a week for this course.
9. Suppose you toss a weighted coin 100 times, and get 15 tails and 85 heads. Find the
approximation for the probability of heads on a single toss of this coin. Use the Empirical
Method.
10. Consider the experiment of tossing a fair coin two times. Find the exact probability of
the event that at least one head occurs on this experiment. Use the Classical Method.
11. An urn contains 100 distinct marbles, 30 or which are green and 70 are blue.
a) Consider the experiment of drawing one marble at random from this urn. Does this
experiment yield equally likely outcomes (i.e. equally likely simple events)? Explain why
or why not.
b) Use your conclusion from part a) to find the probability that a randomly selected
marble from this urn is blue using the Classical Method.
c) Suppose two marbles are drawn from this urn with replacement. What is the
probability that both of them are blue?
12. Suppose A and B are events where P( A)  0.6, P( B)  0.7, P( A and B)  0.42 . a) Find
P( AC ). b) Find P( A or B) . c) Find P( A | B) . d) Find P(B|A). e) Are the events A and
B disjoint? Explain. f) Are the events A and B independent? Explain.
13. There are 25 iPods in a box. Three of them are red and 22 are white. What is the
probability that when two iPods are drawn at random from this box (in a row)
a) both of them will be red? b) at least one of them will be red? c) exactly one will be
red? Note: unless it is otherwise stated, we’ll assume the drawing takes place without
replacement.
14. Suppose that 33.7% of U.S. households have annual income of $75,000 or more.
19.1% of households are in the Northeast region. Also, when a U.S. household is
randomly selected from the Northeast region, the probability that it earned more than
$75,000 is 23.0%.
a) Find the probability that a randomly selected household “earns more than $75,000 and
is from the Northeast”.
b) What proportion of households “earns more than $75,000 or is from the Northeast”.
c) What is the probability that a randomly selected household is from the Northeast given
that it earns more than $75,000?
d) Are the events “a household earns more than $75,000” and “a household is from the
Northeast” independent? Justify.
15. How many combination locks can be formed if a lock is to consist of two letters
followed by five digits?
16. How many ways are there to select a committee of 7 people out of 10 people
available? Do not use statistical commands of your graphing calculator.
17. Suppose a lottery has balls numbered 1 through 50. To win, you must match the
numbers of the three winning balls, not necessarily in the correct order. What is the
probability that you win this lottery by playing it once? Do not use statistical features of
your calculator.
18. Suppose there are 55 Republicans and 45 Democrats at the U.S. Senate. What is the
probability that a randomly selected committee of 6 senators will have 4 Republicans?
19. Consider the experiment of tossing a fair coin two times and observing the resulting
face.
a) What is the set representing the sample space in this experiment?
b) Let X= the discrete random variable representing the number of heads in this
experiment. Construct the probability distribution of X.
c) Construct a probability histogram for X.
d) Compute the mean and standard deviation for X using methods of section 6.1. Do not
use statistical features of your graphing calculator.
e) Interpret your answer obtained for the mean.
xi
20. Consider the table
P( X  xi )
-2
0
1.5 5
0.10 0.20 0.40 0.30
a) Verify whether it is a valid probability distribution of a discrete random variable X.
b) Find P( X  0) . Also, find P( X  0) .
c) Find the mean and standard deviation for X. Do not use statistical features of your
graphing calculator.
21. In the general population, 20% of people are smokers. A group of 30 people is
randomly selected.
a) Describe the probability distribution of a variable that denotes the number smokers in
the 30 people selected. (i.e. give the distribution kind and two parameter values.)
b) Without using statistical features of your graphing calculator, find the probability that
3 of the 30 selected will be smokers.
c) Find the probability that exactly 10 of the 30 selected will be smokers.
d) Find the probability that between 5 and 14, both inclusive, will be smokers.
e) Find the probability that more than 6 will be smokers.
f) Find the probability that less than 10 will be smokers.
g) Compute the mean and standard deviation of the variable that counts the number of
smokers in 30 randomly selected people.
22. The time X of a certain type of a chemical reaction, measured in minutes, has the
probability density function f ( x)  12 x with the domain 0  X  2 .
a) Draw the graph of this probability density function. Hint: First find two points through
which the line y  12 x passes.
b) Verify that this is a valid probability density function of a continuous random variable.
c) Find the probability that a randomly selected chemical reaction of this kind lasts
exactly 1 minute.
d) What proportion of reactions of this kind last at most 1 minute?
e) What is the probability that a randomly selected reaction of this kind will last less than
one minute?
23. Let X be the actual height, in inches, of an adult female. X is approximately normally
distributed with mean of 64.1 inches and standard deviation of 3.8 inches.
a) Compute the probability that a randomly selected adult female is shorter than 60
inches.
b) What proportion of adult females is between 60 and 66 inches tall?
c) If 75% of women are shorter than Alice, what is Alice’s height?
d) If 40% of women are taller than Liz, what is Liz’s height?
e) Find and interpret the 80th percentile of this distribution.
24. Suppose that 65% of all students have a severe headache after taking a math exam. A
specific group of 200 students is studied after a math exam.
a) Can the normal distribution be used to approximate the probability distribution in this
problem? Why?
b) If the normal approximation is used to approximate the variable that counts the number
of students with a headache in the 200 selected, what is the mean and standard deviation
of the resulting normal probability distribution?
c) Use the normal approximation to compute the probability that the number of students
with a headache is between 133 and 143, with both interval endpoints included.
25. Suppose that the mean age of a college student is 20 years old with standard deviation
of 5 years.
a) Describe the sampling distribution of the sample mean that measures the mean age in a
sample of 80 students. Make sure to describe the distribution type, give a reason for it,
give the mean and standard deviation for it.
b) If a random sample of 80 college students is drawn, what is the probability that its
sample mean age is more than 21?
26. According to a CNN report, 7% of the population do not have traditional phones and
instead rely on only cell phones. Suppose a random sample of 200 telephone users is
obtained.
a) Describe the sampling distribution of the sample proportion of telephone users who are
“cell phone users only”.
b) In a random sample of 200 telephone users, what is the probability that more than 8%
are “cell phone users only”?
27. A study of 74 patients with ulcers was conducted in which they were prescribed 40mg
of Pepcid. After 8 weeks, 58 reported healing.
a) Obtain a point estimate for the proportion of patients with ulcers receiving Pepcid who
will report ulcer healing.
b) Verify that the requirements for constructing a confidence interval about p̂ are
satisfied.
c) Construct and interpret a 99% confidence interval for the proportion of patients with
ulcers receiving Pepcid who will report ulcer healing.
28. * Based on a random sample of 1120 Americans 15 years old or older, the sample
mean amount of time spent eating or drinking each day is 1.21 hours with sample
standard deviation of 0.65 hours.
a) Give a point estimate for estimating the mean amount of time an American 15 years or
older spends eating or drinking per day.
b) Verify that the requirements for constructing a confidence interval estimating a
population mean using standard normal distribution are satisfied.
c) Construct and interpret a 90% confidence interval for the mean amount of time spent
eating or drinking each day by an person 15 years old or older living in the U.S. Do not
use the “Z-interval” feature of your graphing calculator.
29. * Now suppose that in a random sample of 44 Americans 15 years old or older, the
sample mean amount of time spent eating or drinking each day is 1.21 hours with sample
standard deviation of 0.65 hours.
a) Give a point estimate for estimating the mean amount of time an American 15 years or
older spends eating or drinking per day.
b) Verify that the requirements for constructing a confidence interval estimating a
population mean are satisfied.
c) Construct and interpret a 90% confidence interval for the mean amount of time spent
eating or drinking each day by an person 15 years old or older living in the U.S. Do not
use the “T-interval” feature of your graphing calculator.
30. In December 2001, 38% of adults with children under the age of 18 reported that their
family ate dinner together every night of the week. Suppose in a recent poll, 446 of 1122
adults with children under the age of 18 reported that their family ate dinner together
every night of the week. Is there sufficient evidence to claim at 10% significance level
that the proportion of adults with children under the age of 18 who ate dinner together
every night of the week has increased recently? Do not use the “1-PropZTest” feature of
your graphing calculator.
31. * The mean birth weight of a baby born in 2005 was 3270 grams. Suppose a random
sample of 40 births to mothers who were less than 15 years old results in a sample mean
birth weight of 3080 grams and standard deviation of 615 grams. Is there sufficient
evidence for a statistician to claim that mothers less than 15 years of age do have lower
birth-weight babies? Use the Classical approach with the significance level of 10%.
Do not use the “T-test” feature of your graphing calculator.
32. * In a certain year a school administrator claims that students whose first language is
not English score differently on the math portion of the SAT exam than those whose first
language is English. That year the SAT math scores of students whose first language is
English were approximately normally distributed with the mean of 516.
A sample of 20 students whose first language is not English is randomly selected in
which the mean is 523 and standard deviation 114. Is there sufficient evidence that
supports the administrator’s claim? Use the P-value approach with the significance level
of 1%. Do not use the “T-test” feature of your graphing calculator.
Note: Problems above marked with a * symbol denote the most important problems for the course.
----------------------------------------------- Short answers ----------------------------------------1. a)
X=time spent watching TV Frequency Relative Frequency
3
[4,6)
3
8
[6,8)
2
2
8
[8,10)
[10,12)
0
3
0
Total
8
1
3
8
b) Not shown. c) Quantitative continuous.
2. a) x  2 . b) s 2  463 . c) s  463
a)   15 b)   5 2
a) i.
b) ii.
30
xw  13
The woman.
a) P60  70.5 b) Q1  67 c) M  70
a) The number of study hours – independent variable. b) Not shown. The
relationship appears linear. c) r  0.96 strong, positive correlation. d)
y  6.86 x  37.62 . Graph not shown. e) The test grade is 71.92.
85
9. P( H )  100
.
10. P(at least one head)  34 .
70
11. a) Yes because it’s a random drawing. b) P( B)  100
. c) P(B 1 and B 2 )= 0.49.
3.
4.
5.
6.
7.
8.
12. a) P( AC )  0.4 . b) P(A or B)= 0.88. c) P(A | B)= 0.6 d) P(B | A)= 0.7
e) No, since P(A and B)= 0.42  0 , so “A and B” is not the empty set.
f) Yes. Since P(A | B)= P(A), as 0.6=0.6.
6
132
13. a) P( R1 and R2 )  600
. b) P(at least one R)  138
600 . c) P(one R)= 600
14. a) P(M and N)  0.0439. b) P(M or N)  0.4841. c) P(N|M)  0.1303
d) No. P( N | M )  P( N ) , as 0.1303  0.1901.
15. 67,600,000 locks.
16. 120 ways.
17. P( winning )  19,1600
18. P(4R and 2D)  0.2832.
19. a) {HH, HT, TH, TT} b) See the table below. c) Not shown. d)   1 ,  
e) Not shown.
xi P( X  xi )
1
0
4
1
2
4
2
1
4
20. a) i. 0  P( X  xi )  1 for all xi (justify it) and ii.
1
2
.
k
 P( X  x )  1 (justify it). Yes,
i 1
i
valid. b) P( X  0)  20%, P( X  0)  90%. c)   1.9 ,   5.19 .
21. a) Binomial( n  30, p  0.20). b) P( X  3)  0.0785. c) P( X  10)  0.0355 .
d) P(5  X  14)  0.7446. e) P( X  6)  0.3930. f) P( X  10)  0.9389.
22.
23.
24.
25.
g)   6 ,   4.8 .
a) Not shown. b) i. f ( x)  0 for all x (justify it) and ii. The area under the graph of f
is 1 (justify it). Yes, valid. c) P( X  1)  0 . d) P( X  1)  14 e) P( X  1)  14 .
a) P( X  60)  0.1403. b) P(60  X  66)  0.5512. c) x  66.66. d) x  65.06
e) x  67.30.
a) Yes, since np(1  p)  10 b)   130 ,   45.5 .
c) P(133  X  143)  0.3328.
a) Since n  80  30, by CLT, X is approximately normal with  X  20 , and
X 
5
80
.
b) P( X  21)  0.0368.
26. a) p̂ is approximately normal with  pˆ  0.07 ,  pˆ  0.02 . b) P( pˆ  8%)  0.3085 .
27. a) pˆ1  0.78. b) n( pˆ1 )(1  pˆ1 )  10 and n  5%  (large population size).
c) [0.66,0.91]
28. a) x1  1.21 b) n  1120  30 and n  1000 c) [1.18, 1.24].
29. a) x1  1.21 b) n  44  30 and n  5%  (large population size) c) [1.04, 1.38].
30. P-value  11.31%, but   10%. Thus, do not reject H 0 .
31. t0  1.95 and rejection region is (,1.304] . Thus, reject H 0 .
32. P-value  79.00%, but   1%. Thus, do not reject H 0 .