(Penn State University) Stat 500 Midterm 1 Latest
Update 2025-2026 Actual Exam 120 Questions
with 100% Verified Correct Answers Guaranteed
A+ Verified by Professor
Adding by a constant - CORRECT ANSWER: Adding a Constant: Suppose students in
one of my classes did relay bad in their first midterm and I decided to curve the grades
by adding 5 points to everyone.
New mean = old mean + 5
New median = old median + 5
New SD = same
New IQR = same
New variance = same
Bayes' Theorem - CORRECT ANSWER: Way to find the conditional probability of an
event P(A | B), when the "reverse" conditional probability P(B | A) is the probability that
is known. Need P(A), P(B)>0.
Bayes' Theorem Example - CORRECT ANSWER: A test indicates the presence of a
particular disease 90% of the time when the disease is present, and the presence of the
disease is 2% when the disease is not present. If 0.5% of the population has the
disease, calculate the conditional probability that a person selected at random has the
disease if the test indicates the presence of the disease.
P(+) = test is positive
P(-) = test is negative
D = have the disease
D' = don't have the disease
,P(AnB) = P(A|B)*P(B)
P(+|D) = 0.9
P(+|D') = 0.02
P(-|D') = 0.98
P(D) = 0.005 --> P(D') = 0.995
P(D|+) = P(Dn+)/P(+) = P(+|D)*P(D)/P(+)
P(+|D)*P(D)/P(Dn+)+P(D'n+)
P(+|D)*P(D)^ ^P(+|D')*P(D')
0.9*0.005/(0.9*0.005)+(0.02*0.995) = 0.184=18.4%
P(D|+) = 18.4%
Bernoulli Distribution - CORRECT ANSWER: The probability distribution of a random
variable with two possible outcomes, each with a constant probability of occurrence.
Example: heads and tails
Binomial Distribution - CORRECT ANSWER: When there is a sequence of Bernoulli
trials, we are often interested in the total number of successes, not the order of their
occurrence.
Binomial Distribution Conditions - CORRECT ANSWER: For X to have a binomial
distribution these conditions need to be satisfied:
o Fixed number of trials
o Each trial must have two possible outcomes
, o Trails are independent
o Probability of success remains the same from trial to trial.
Binomial Distribution Equation - CORRECT ANSWER: Denoted by:
X~Bin(N,P)
Equation:
Binomial Distribution Example - CORRECT ANSWER: Example 1: Suppose I toss a
coin 7 times. If X is the number of heads I get, what is the probability that I get exactly 6
heads?
X~Bin(7,0.5)
P(X=6) = (7Choose6)0.5^6 * 0.5^1 = 0.055
Categorical Variables - CORRECT ANSWER: Binary: two possible outcomes (male and
female)
Ordinal: There is a natural ordering to the levels (Education level)
Nominal: when there is no intrinsic ordering among levels (eye colour, ethnicity)
Central Limit Theorem (CLT) - CORRECT ANSWER: the sampling distribution derived
from a simple random sample that is sufficiently big enough will be approximately
normally distributed (N = 30)
characteristics of a normal distribution - CORRECT ANSWER: o Always on or above the
horizontal line
o Total area under the curve is 1
o The 𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) is given by the area under the curve between a and b.
· Mean is located at the center of the symmetric curve. Changing 𝜇 without changing 𝜎 ,
moves the normal curve along the horizontal axis without changing it's spread.
Update 2025-2026 Actual Exam 120 Questions
with 100% Verified Correct Answers Guaranteed
A+ Verified by Professor
Adding by a constant - CORRECT ANSWER: Adding a Constant: Suppose students in
one of my classes did relay bad in their first midterm and I decided to curve the grades
by adding 5 points to everyone.
New mean = old mean + 5
New median = old median + 5
New SD = same
New IQR = same
New variance = same
Bayes' Theorem - CORRECT ANSWER: Way to find the conditional probability of an
event P(A | B), when the "reverse" conditional probability P(B | A) is the probability that
is known. Need P(A), P(B)>0.
Bayes' Theorem Example - CORRECT ANSWER: A test indicates the presence of a
particular disease 90% of the time when the disease is present, and the presence of the
disease is 2% when the disease is not present. If 0.5% of the population has the
disease, calculate the conditional probability that a person selected at random has the
disease if the test indicates the presence of the disease.
P(+) = test is positive
P(-) = test is negative
D = have the disease
D' = don't have the disease
,P(AnB) = P(A|B)*P(B)
P(+|D) = 0.9
P(+|D') = 0.02
P(-|D') = 0.98
P(D) = 0.005 --> P(D') = 0.995
P(D|+) = P(Dn+)/P(+) = P(+|D)*P(D)/P(+)
P(+|D)*P(D)/P(Dn+)+P(D'n+)
P(+|D)*P(D)^ ^P(+|D')*P(D')
0.9*0.005/(0.9*0.005)+(0.02*0.995) = 0.184=18.4%
P(D|+) = 18.4%
Bernoulli Distribution - CORRECT ANSWER: The probability distribution of a random
variable with two possible outcomes, each with a constant probability of occurrence.
Example: heads and tails
Binomial Distribution - CORRECT ANSWER: When there is a sequence of Bernoulli
trials, we are often interested in the total number of successes, not the order of their
occurrence.
Binomial Distribution Conditions - CORRECT ANSWER: For X to have a binomial
distribution these conditions need to be satisfied:
o Fixed number of trials
o Each trial must have two possible outcomes
, o Trails are independent
o Probability of success remains the same from trial to trial.
Binomial Distribution Equation - CORRECT ANSWER: Denoted by:
X~Bin(N,P)
Equation:
Binomial Distribution Example - CORRECT ANSWER: Example 1: Suppose I toss a
coin 7 times. If X is the number of heads I get, what is the probability that I get exactly 6
heads?
X~Bin(7,0.5)
P(X=6) = (7Choose6)0.5^6 * 0.5^1 = 0.055
Categorical Variables - CORRECT ANSWER: Binary: two possible outcomes (male and
female)
Ordinal: There is a natural ordering to the levels (Education level)
Nominal: when there is no intrinsic ordering among levels (eye colour, ethnicity)
Central Limit Theorem (CLT) - CORRECT ANSWER: the sampling distribution derived
from a simple random sample that is sufficiently big enough will be approximately
normally distributed (N = 30)
characteristics of a normal distribution - CORRECT ANSWER: o Always on or above the
horizontal line
o Total area under the curve is 1
o The 𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) is given by the area under the curve between a and b.
· Mean is located at the center of the symmetric curve. Changing 𝜇 without changing 𝜎 ,
moves the normal curve along the horizontal axis without changing it's spread.
