AMAT 108 Elementary Statistics
Spring 2024
Exam 2
Version A
Print Name:
UAlbany Email:
Instructor’s Name and Section:
Directions: You have 80 minutes to answer the following questions. You must show all
necessary work as neatly and clearly as possible and clearly indicate your final answers.
No notes, textbooks, mobile phones or other aids are allowed. Only scientific calculators are allowed.
Do not detach these pages.
Question
Possible Points Points Earned
Multiple Choice 1-6
6
Free Response 7
10
Free Response 8
8
Free Response 9
7
Free Response 10
10
Free Response 11
10
Free Response 12
8
Free Response 13
5
Total Points
64
Percentage
PART A: MULTIPLE-CHOICE QUESTIONS (1 PT. EACH)
Directions: In this part, you will choose ONE answer from among the choices given. No explanation
is required to be shown and no partial credit will be given for the multiple choice questions.
1. The following sample space is the color of the post-it notes (in the order they were used) from a
random sample of three consecutive post-it notes were needed where Y = yellow and B = blue:
S = {Y Y Y, Y Y B, Y BY, BY Y, Y BB, BY B, BBY, BBB}
If event Z represents at least one of the three post-it notes is blue, then which of the following
is the list of outcomes for event Z?
(A) Z = {Y Y B, Y BY, BY Y }
(D) Z = {Y Y Y, Y Y B, Y BY, BY Y }
(B) Z = {Y Y Y }
(E) Z = {Y BB, BY B, BBY, BBB}
(C) Z = {Y Y B, Y BY, BY Y, Y BB, BY B, BBY, BBB}
2. The following distribution table give the probabilities of the food that was ordered during a
random week at a restaurant. Note: The probabilities are disjoint of each other (no overlap).]
Food
Chicken Veal Pizza Steak Pasta Fish
Probability 0.250
0.129 0.295 0.117 0.097
What is the probability that the food ordered during the random week at the restaurant was veal?
(A) 0.167
(D) 0.112
(B) 0.888
(E) 0.178
(C) 0
3. Using the same distribution table in Question 2, what is the probability that the food ordered
during the random week at the restaurant was pizza or pasta?
(A) 0.246
(B) 0.041
(C) 0.012
(D) 0.015
(E) 0.123
4. Using the standard normal curve, find P (−2.98 ≤ z ≤ −0.44).
(A) 0.0055
(D) 2.54
(B) 0.3286
(E) 0.9945
(C) 0.6714
5. Which of the following z-scores best represents the top 13% of all z-score values?
(A) 0.5517
(D) −2.23
(B) 2.23
(E) 1.13
(C) −1.13
6. An 80% confidence interval for the population proportion based on a random sample is (0.375, 0.419).
Which of the following is the correct interpretation of the confidence interval?
(A) We are confident that the population proportion will be within this interval 80% of the time.
(B) We are confident that 80% of the population proportion will be within this interval.
(C) We are 80% confident that the population proportion will be within this interval.
(D) There is an 80% probability that the population proportion will be within this interval.
PART B: FREE RESPONSE QUESTIONS
Directions: For questions 7-13, you must show ALL necessary work to receive full credit.
An answer with no work, even if correct, will not receive full credit. Please circle or box your final
answer.
7. According to a nationwide survey in 2010, the probability of a part-time college student working
at a job full-time (35+ hours) is 0.43. Looking at a random sample of four part-time college
students and having each student’s result being independent of each other, find the following:
(a) What is the probability that all four of them are working at a job full-time? Round your
answer to three decimal places. (3 pts.)
(b) What is the probability that the first two students work at a job full-time but the last two
students don’t? Round your answer to three decimal places. (3 pts.)
(c) What is the probability that at least one of the students work at a job full-time? Round
your answer to three decimal places. (4 pts.)
8. A probability distribution of a discrete random variable is given below:
4
8
13
x
Probability 0.30 0.56 0.14
(a) Find the mean/expected value (µx ) of the discrete random variable. Round your answer
to two decimal places, if needed. (3 pts.)
(b) Find the standard deviation (σx ) of the discrete random variable. ONLY ROUND your
answer to three decimal places. (5 pts.)
9. Suppose a continuous random variable has a uniform distribution on the interval from x = 29 to
x = 65. Find the following:
(a) The height of the uniform distribution. Either leave your answer in fraction form or
round your answer to three decimal places. (2 pts.)
(b) P (x ≤ 38). Round your answer to three decimal places. (3 pts.)
(c) P (47 < x < 60). Round your answer to three decimal places. (2 pts.)
10. The amount of meat that a US person consumes per year is normally distributed with a mean
of 224.57 pounds and a standard deviation of 18.23 pounds.
(a) What is the probability that a single US person chosen at random from all people in the
US consumed 230 pounds of meat? Hint: Find P (x ≥ 230). (4 pts.)
(b) A random sample of 97 US people were selected. What are the mean and standard deviation
of the sampling distribution of x? Round the standard deviation to five decimal
places. (3 pts.)
(c) What is the probability that the average amount of meat consumed from the 97 US people
is 230 pounds or higher? Hint: Find P (x ≥ 230). (3 pts.)
11. In 2023, roughly 66% of Americans live in their own homes. A random sample of 129 US households were selected from the population.
(a) Show that the sampling distribution of p̂ is approximately normal using the success-failure
conditions. (3 pts.)
(b) Based on the random sample of 129 US households, what are the mean and the standard
deviation of the sampling distribution of p̂? Round the standard deviation to five
decimal places. (3 pts.)
(c) Based on the random sample of 129 US households, what is the probability that less than
60% of Americans live in their own homes? Hint: Find P (p̂ < 0.60). (4 pts.)
12. From a random sample of 480 Americans, approximately 32.5% of them said they read printed
books (rather than reading digital books).
(a) Construct a 95% confidence interval for the population proportion, p, of all Americans
reading printed books. Round all work and interval endpoints to three decimal
places. (5 pts.)
(b) A report estimated that the proportion of all Americans reading printed books is p = 0.35.
Using this estimate, determine the sample size required to be within a margin of error
(bound on the error of estimation) of 0.025 with a 95% confidence interval. Round your
answer UP to the nearest whole number. (3 pts.)
13. From a random sample of 14 standard-size candy bars, the average number of calories was
243.21 calories with a standard deviation of 23.8 calories. Assume the distribution of the
number of calories from all standard-size candy bars is normal.
(a) Find the margin of error (bound on the error of estimation) for a 90% confidence level for
µ, the mean number of calories for all standard-size candy bars. If you are calculating
the standard error by itself first, then make sure it goes to five decimal places.
Otherwise, round your answer to two decimal places. (3 pts.)
(b) Construct the 90% confidence interval for µ. (2 pts.)
Formula Sheet
• Probability:
Mutually exclusive or disjoint events: P (A or B) = P (A ∪ B) = P (A) + P (B)
Independent Events: P (A and B) = P (A ∩ B) = P (A)P (B)
• Probability of the complement of an event: P (not A) = P (AC ) = 1 − P (A)
• At least one Rule: P (at least one success in n trials) = 1 − P (no successes in n trials)
• Finding the height of a Uniform Distribution:
height =
1
base
• Probability (area) of a Uniform Distribution:
Probability = base × height
• Probabilities of continuous random variables:
P (a ≤ x ≤ b) = P (a < x ≤ b) = P (a ≤ x < b) = P (a < x < b)
• The probability between two endpoints of any continuous random variable:
P (a ≤ x ≤ b) = P (x ≤ b) − P (x ≤ a)
P (a ≤ z ≤ b) = P (z ≤ b) − P (z ≤ a)
• Mean (expected value) of a discrete random variable:
X
xp(x)
µx = E(x) =
x
• Standard deviation of a discrete random variable:
s
X
p
σx = E(x2 ) − (E(x))2 =
(x − µx )2 p(x)
x
• Mean and standard deviation for sampling distribution of x:
σ
St. deviation = σx = √
n
where µ = population mean and σ = population standard deviation
Mean value = µx = µ
• z-score:
z=
x−µ
σ
• Standardized variable (z-score) for x when σ known:
x−µ
z= σ
√
n
• Mean and standard deviation for sampling distribution of p̂:
r
Mean value = µp̂ = p
St. deviation = σp̂ =
p(1 − p)
n
where p = population proportion of successes and
number of successes in sample
= sample proportion of successes
p̂ =
n
• Standardized variable (z-score) for p̂:
z=r
p̂ − p
p(1 − p)
n
• Success-failure conditions for sampling distribution of p̂ to be approximately normal:
np ≥ 10 and n(1 − p) ≥ 10
• Confidence interval form:
point estimate ± margin of error(bound on error of estimation)
• Confidence interval for a population proportion:
r
r
p̂(1 − p̂)
p̂(1 − p̂)
p̂ ± zcrit
= p̂ ± z ∗
n
n
• Determining the sample size needed to be within B = margin error (bound on the error)
under a certain confidence level:
∗ 2
z 2
z
crit
n = p(1 − p)
= p(1 − p)
B
B
• To find the conservatively large sample size needed, set p = 0.5.
• Confidence interval for a population mean when σ unknown:
s
t∗ s
x ± tcrit √ = x ± √
n
n
df = n − 1