Binomial and Normal Distributions: Online Learning Quizzes

Question 1

What is a defining characteristic of the binomial distribution?

Accepted Answer

Models the number of successes in a sequence of independent Bernoulli trials

Answer

Always symmetric about the mean

Answer

Variance is always less than its mean

Question 2

Which of the following is NOT a characteristic of a binomial distribution?

Accepted Answer

It describes the number of trials until a success occurs.

Answer

The probability of success remains constant for each trial.

Answer

The number of trials is fixed.

Answer

The outcome of each trial is independent.

Question 3

Which of the following is the correct formula for the variance of a binomial distribution?

Accepted Answer

σ^2 = npq

Answer

σ^2 = np

Answer

σ^2 = nq

Answer

σ^2 = √(npq)

Question 4

A factory produces light bulbs with a 5% defective rate. If 100 bulbs are randomly selected, what is the probability that at least 90 are not defective?

Accepted Answer

0.9938

Answer

0.0062

Answer

0.05

Answer

0.95

Question 5

Which of the following is the shape of a normal distribution?

Accepted Answer

Symmetrical and bell-shaped

Answer

Lopsided and skewed

Answer

Uniform and flat

Answer

Triangular

Question 6

What is the probability of randomly selecting a score between 70 and 80 from a normal distribution with mean 75 and standard deviation 5?

Accepted Answer

0.3520

Answer

0.2420

Answer

0.1121

Answer

0.1750

Question 7

Which distribution would be most appropriate to model the time it takes for a customer to be served at a fast-food restaurant?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Question 8

In a normal distribution, approximately what percentage of data values fall within one standard deviation of the mean?

Accepted Answer

68.26%

Answer

34.13%

Answer

95.44%

Answer

99.73%

Question 9

The variance of a normal distribution is a measure of its:

Accepted Answer

Spread or variability

Answer

Central tendency

Answer

Skewness

Answer

Kurtosis

Question 10

Which of the following expressions represents the probability density function of a normal distribution?

Accepted Answer

f(x) = (1 / (σ√(2π))) * e^(-(x - μ)² / (2σ²))

Answer

f(x) = (1 / (σ²√(2π))) * e^(-(x - μ) / (2σ))

Answer

f(x) = (1 / (σ√(2π))) * e^(-(x² - μ²) / (2σ²))

Answer

f(x) = (1 / (σ²√(2π))) * e^(-(x - μ)²)

Question 11

If X and Y are independent binomial random variables with parameters (n₁, p₁) and (n₂, p₂) respectively, what are the parameters of their sum, Z = X + Y?

Accepted Answer

(n₁ + n₂, p₁ + p₂)

Answer

(n₁ + n₂, p₁ * p₂)

Answer

(n₁ - n₂, p₁ - p₂)

Answer

(n₁ - n₂, p₁ * p₂)

Question 12

The time required to complete a task follows an exponential distribution with a mean of 10 minutes. What is the probability that it takes more than 15 minutes to complete the task?

Accepted Answer

0.0014

Answer

0.0003

Answer

0.0067

Answer

0.0228

Question 13

Which of the following is the probability mass function (PMF) of the binomial distribution?

Accepted Answer

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Answer

P(X = k) = (n choose k) * k * p^k * (1-p)^(n-k)

Answer

P(X = k) = (n choose k) * p^k * (1-p)^(n+k)

Answer

P(X = k) = (n choose k) * (1-p)^k * (p)^(n-k)

Question 14

What is the expected value (mean) of a binomial distribution with parameters n and p?

Accepted Answer

np

Answer

np/2

Answer

n(1-p)

Answer

np(1-p)

Question 15

What is the characteristic shape of a normal distribution called?

Accepted Answer

Bell-shaped

Answer

U-shaped

Answer

Triangular

Answer

Exponential

Question 16

Which function describes the role of the standard deviation in a normal distribution?

Accepted Answer

It determines the spread or dispersion of data points.

Answer

It calculates the mean or central tendency of data points.

Answer

It indicates the probability of a specific data point occurring.

Answer

It has no significant impact on the distribution.

Question 17

Which formula calculates the z-score for a value x in a normal distribution with mean μ and standard deviation σ?

Accepted Answer

(x - μ) / σ

Answer

(μ - x) / σ

Answer

σ / (x - μ)

Answer

μ / (σ - x)

Question 18

According to the empirical rule for normal distributions, what percentage of data falls within one standard deviation of the mean?

Accepted Answer

68%

Answer

34%

Answer

95%

Answer

99.7%

Question 19

Which scenario best exemplifies an application of the binomial distribution?

Accepted Answer

Counting the number of heads obtained in 10 coin flips.

Answer

Measuring the heights of a group of people.

Answer

Determining the average temperature over a month.

Answer

Analyzing the distribution of test scores.

Question 20

Which distribution is commonly used to model continuous data?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Geometric distribution

Question 21

How do a population and a sample differ?

Accepted Answer

A population is the complete set of individuals, while a sample is a subset used for research.

Answer

A sample is the complete set of individuals, while a population is a subset used for research.

Answer

A population represents theoretical data, while a sample represents real-world data.

Answer

There is no distinction between a population and a sample.

Question 22

In a binomial experiment with **n** trials and probability of success **p**, what is the formula for calculating the mean, representing the expected number of successes?

Accepted Answer

Mean = n * p

Answer

Mean = n / p

Answer

Mean = p / n

Answer

Mean = np^2

Question 23

Which of the following is a key characteristic of a binomial distribution?

Accepted Answer

It can only take integer values

Answer

It is symmetric about the mean

Answer

Its mean is always less than the median

Answer

Its mode is always equal to the mean

Question 24

Which of the following is the formula for calculating the variance of a normal distribution?

Accepted Answer

Variance = σ^2

Answer

Variance = σ

Answer

Variance = μ

Answer

Variance = μ^2

Question 25

In a normal distribution, the area under the curve between any two z-scores represents:

Accepted Answer

The probability of a value falling within that range

Answer

The mean of the distribution

Answer

The standard deviation of the distribution

Answer

The variance of the distribution

Question 26

Which of the following is **not** a key difference between a binomial distribution and a normal distribution?

Accepted Answer

Normal distribution can be bimodal, while binomial distribution cannot

Answer

Binomial distribution takes discrete values, while normal distribution takes continuous values

Answer

Normal distribution is symmetric, while binomial distribution can be skewed

Answer

Binomial distribution has a finite range, while normal distribution has an infinite range

Question 27

In a binomial distribution, what symbol represents the probability of success on each trial?

Accepted Answer

p

Answer

q

Answer

n

Answer

μ

Question 28

What represents the mean of a binomial distribution with parameters n and p?

Accepted Answer

np

Answer

nq

Answer

np(1-p)

Answer

n(1-p)

Question 29

Which formula expresses the probability mass function of a binomial distribution?

Accepted Answer

P(X = k) = (n choose k) * p^k * q^(n-k)

Answer

P(X = k) = (n choose k) * q^k * p^(n-k)

Answer

P(X = k) = k * p^k * q^(n-k)

Answer

P(X = k) = n * p^k * q^(n-k)

Question 30

Which formula represents the probability density function of a normal distribution with mean μ and standard deviation σ?

Accepted Answer

f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)^2 / (2σ^2))

Answer

f(x) = (1 / (σ^2 * √(2π))) * e^(-(x - μ)^2 / (2σ))

Answer

f(x) = (1 / (2π)) * e^(-(x - μ)^2 / (2σ^2))

Answer

f(x) = (1 / (2σ)) * e^(-(x - μ)^2 / (2σ^2))

Question 31

What represents the standard deviation of a normal distribution with mean μ?

Accepted Answer

σ

Answer

μ

Answer

σ^2

Answer

μ^2

Question 32

What formula calculates the z-score of a data point x in a normal distribution with mean μ and standard deviation σ?

Accepted Answer

(x - μ) / σ

Answer

(x + μ) / σ

Answer

(x - σ) / μ

Answer

(x + σ) / μ

Question 33

What is the area under the normal curve between z-scores -1 and 1?

Accepted Answer

0.6827

Answer

0.3413

Answer

0.8413

Answer

0.9545

Question 34

Which of the following is the probability mass function of a binomial distribution with parameters (n) and (p)?

Accepted Answer

P(X = k) = {n choose k} * p^k * (1-p)^(n-k)

Answer

P(X = k) = n * p^k * (1-p)^(n-k)

Answer

P(X = k) = {n choose k} * p^k * q^(n-k)

Answer

P(X = k) = n * p^k

Question 35

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is bimodal.

Answer

It is symmetric.

Answer

It is continuous.

Answer

It is bell-shaped.

Question 36

If (X) follows a normal distribution with mean 50 and standard deviation 10, what is the probability that (X) is less than 40?

Accepted Answer

0.0038

Answer

0.1587

Answer

0.9962

Answer

0.5000

Question 37

Which of the following is the expected value of a binomial distribution with parameters (n) and (p)?

Accepted Answer

np

Answer

(1 - p)/p

Answer

n/p

Answer

p/(1 - p)

Question 38

Which of the following is the mode of a normal distribution?

Accepted Answer

Mean

Answer

Standard deviation

Answer

Median

Answer

Variance

Question 39

If (X) follows a binomial distribution with (n = 5) and (p = 0.6), what is the standard deviation?

Accepted Answer

√(5 × 0.6 × 0.4)

Answer

√(5 × 0.6)

Answer

√(0.6 × 0.4)

Answer

√(5 × 0.4)

Question 40

Which of the following represents the cumulative distribution function of the normal distribution with parameters μ and σ?

Accepted Answer

P(X ≤ x) = (1 / √(2πσ^2)) ∫(-∞,x) e^(-(x - μ)^2 / (2σ^2)) dx

Answer

P(X ≤ x) = (1 / √(2πσ^2)) ∫(x,∞) e^(-(x - μ)^2 / (2σ^2)) dx

Answer

P(X ≤ x) = (1 / σ) ∫(-∞,x) e^(-(x - μ)^2 / (2σ^2)) dx

Question 41

If (X) follows a normal distribution with mean 100 and standard deviation 20, what is the probability that (X) is between 80 and 120?

Accepted Answer

0.9544

Answer

0.5000

Answer

1.0000

Answer

0.0456

Question 42

A multiple-choice test has 20 questions, each with 4 options. If a student guesses on each question, what is the probability of getting exactly 10 questions correct?

Accepted Answer

0.0816

Answer

0.0625

Answer

0.1250

Answer

0.2500

Question 43

The number of defective items in a batch of 100 items follows a binomial distribution with a probability of success of 0.1. What is the expected number of defective items?

Accepted Answer

10

Answer

15

Answer

20

Answer

5

Question 44

The heights of adult males are normally distributed with a mean of 175 cm and a standard deviation of 10 cm. What is the probability of randomly selecting an adult male who is taller than 185 cm?

Accepted Answer

0.1587

Answer

0.3413

Answer

0.0668

Question 45

Which of the following is NOT a property of the normal distribution?

Accepted Answer

Symmetric but not unimodal

Answer

Bell-shaped

Answer

Mean, median, and mode are equal

Answer

Asymptotic tails

Question 46

Which of the following measures of central tendency is most appropriate for a skewed distribution?

Accepted Answer

Median

Answer

Mean

Answer

Range

Answer

Mode

Question 47

A continuous random variable X has a probability density function f(x) = (1/3)x^2 for 0 < x < 1. What is the cumulative distribution function of X?

Accepted Answer

F(x) = (1/3)x^3

Answer

F(x) = x^2

Answer

F(x) = (1/3)x

Question 48

Which of the following transformations does NOT affect the mean of a normal distribution?

Accepted Answer

Standardization

Answer

Square root transformation

Answer

Logarithmic transformation

Answer

Linear transformation