Question 1

Suppose X is a random variable with the binomial distribution with n = 4 and p = 1/4. The probability that X is greater than or equal to 1 is:
A)0.9961.
B)0.6836.
C)0.3164.
D)0.0039.

Accepted Answer

We can use the formula for the probability mass function of the binomial distribution to calculate the probability:

P(X>=1) = 1 - P(X=0)

P(X=0) = (4 choose 0) * (1/4)^0 * (3/4)^4
= 1 * 1 * 27/256
= 27/256

So, P(X>=1) = 1 - 27/256
= 229/256
≈ 0.6836

Therefore, the correct choice is B.

Question 2

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. Last month, 100 horses were admitted and 20 of them had been infected with West Nile virus. The probability of seeing 20 or more horses with West Nile virus was calculated based on a Normal distribution with $\mu$ = 15 and $\sigma$= 12.75. Which of the following statements is true?
A)Using the Normal approximation to the binomial distribution is justified, because X = 20 > 10.
B)Using the Normal approximation to the binomial distribution is justified, because n is large and exact calculations are too tedious.
C)Using the Normal approximation to the binomial distribution is justified, because np = 15 and n(1 - p)= 85.
D)All of the answer options are correct.

Accepted Answer

Using the Normal approximation to the binomial distribution is justified because n is large (n = 100), np = 15, and n(1 - p) = 85.

Question 3

A local veterinary clinic recently had 25 horses admitted from the same barn. It was determined that most cases of West Nile virus infection in the group of 25 horses were caused by infection by another horse. In this case, modeling the number of cases diagnosed with a binomial distribution is not appropriate because:
A)the success on different trials needs to be mutually exclusive and this is not the case here.
B)the success on individual trials needs to be independent.
C)the success on individual trials needs to be positive.
D)n = 25 is a very small sample.

Accepted Answer

In the case of the spread of West Nile virus infection among the 25 horses, the success on individual trials (horses) is not independent. The infection in one horse can easily spread to another horse, violating the assumption of independence needed for a binomial distribution.

Question 4

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability at least one of the 10 horses has been infected with West Nile virus?
A)0.923
B)0.177
C)0.348
D)0.803

Accepted Answer

The answer of A local veterinary clinic typically sees 15%...

Question 5

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability a randomly selected horse among the 10 new admits has West Nile virus?
A)0.1
B)0.5
C)0.15
D)0.9

Accepted Answer

If 15% of the horses present with West Nile virus, then out of 10 horses randomly admitted, we can expect 10 x 15% = 1.5 to have West Nile virus. However, since we can't have half a horse with West Nile virus, we round to the nearest whole number and expect 1 horse with West Nile virus. Therefore, the probability of selecting a horse with West Nile virus from the 10 new admits is 1/10 or 0.15.

Question 6

Two students taking a multiple choice exam with 20 questions and four choices for each question have the same incorrect answer on seven of eight incorrectly answered problems. The probability that student B guesses the same incorrect answer as student A on a particular question is 1/4. If the student is guessing, the guess for one problem is independent of the guess for the other problems. The probability of getting agreement on seven or more out of eight questions is:
A)0.9996.
B)0.0004.
C)0.25.
D)None of the answer options is correct.

Accepted Answer

The probability that student A and student B both guess the same incorrect answer for a particular question is 1/4. Therefore, the probability that they both guess different answers is 1 - 1/4 = 3/4. Since there are 20 questions, the probability that they have agreement on 7 or more questions is:

${20 \choose 7} (1/4)^7 (3/4)^{13} + {20 \choose 8} (1/4)^8 (3/4)^{12}$

This can be computed using a binomial distribution or a calculator to get approximately 0.0004.

Question 7

Suppose we toss a fair coin repeatedly. We continue to do this until a tail is observed. Let X be the number of tosses required. Then X has:
A)a binomial distribution, with mean 0.5.
B)a binomial distribution, with mean 2.
C)a binomial distribution, with variance 0.707.
D)None of the answer options is correct.

Accepted Answer

X follows a geometric distribution, not a binomial distribution, because it measures the number of trials until the first success (in this case, getting a tail). The mean and variance of a geometric distribution are not given in the options correctly for a fair coin toss scenario.

Question 8

Two students taking a multiple choice exam with 20 questions and four choices for each question have the same incorrect answer on eight of the problems. The probability that student B guesses the same incorrect answer as student A on a particular question is 1/4. If the student is guessing, it is reasonable to assume guesses for different problems are independent. The instructor for the class suspects the students exchanged answers. The teacher decides to present a statistical argument to substantiate the accusation. A possible model for the number of incorrect questions that agree is:
A)a binomial distribution with n = 8 and p = 0.25.
B)a binomial distribution with n = 20 and p = 0.4.
C)a Normal distribution with $\mu$ = 8 and $\sigma$ = 0.25.
D)a Normal distribution with $\mu$ = 20 and $\sigma$ = 0.4.

Accepted Answer

The number of incorrect questions that agree follows a binomial distribution with n = 8 (the number of questions on which they both got the wrong answer) and p = 0.25 (the probability that they both guess the same wrong answer).

Question 9

A college basketball player makes 70% of his free throws. At the end of a game, his team is losing by two points. He is fouled attempting a three-point shot and is awarded three free throws. Assuming each free throw is independent, what is the probability that he makes at least two of the free throws?
A)0.784
B)0.7
C)0.441
D)0.216

Accepted Answer

The probability of making at least two free throws can be calculated by finding the sum of the probabilities of making exactly two and making all three. The probability of making exactly two free throws is 3 * (0.7)^2 * (0.3) = 0.441, and the probability of making all three is (0.7)^3 = 0.343. Adding these together gives 0.441 + 0.343 = 0.784.

Question 10

Opinion polls find that 20% of American adults claim they never have time to relax. Suppose you take a random sample of 200 American adults, and you count the number X in your sample that claim they never have time to relax. Using the Normal approximation, the probability that X is at least 50 is:
A)less than 0.0001.
B)about 0.962.
C)about 0.2.
D)about 0.038.

Accepted Answer

The sample size (n=200) is large enough for us to use the Normal approximation. We can model the number X of individuals who claim they never have time to relax with a normal distribution with mean μ = np and standard deviation σ = sqrt(np(1-p)), where n=200 and p=0.2.
Thus, X ~ N(μ=40, σ=4.00).
To find the probability that X is at least 50, we can standardize X:
Z = (50 - 40)/4.00 = 2.50
And we can find the probability using a standard normal table or calculator as P(Z ≥ 2.50) = 0.0062.
Therefore, the answer is D) about 0.038.

Question 11

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The standard deviation of X is:
A)0.71
B)0.768.
C)0.8.
D)0.88.

Accepted Answer

The answer of At a large Midwestern college, 4% of...

Question 12

Two students taking a multiple choice exam with 20 questions and four choices for each question have the same incorrect answer on seven of eight incorrectly answered problems. The probability that student B guesses the same incorrect answer as student A on a particular question is 1/4. If the student is guessing, the guess for one problem is independent of the guess for the other problems. The number of guesses by student B that agree with those of student A has mean and variance given by: A)$\mu$ = 2 and $\sigma$²= 1.5. B)$\mu$ = 2 and $\sigma$²= 1.5². C)$\mu$= 4 and $\sigma$²= 1.5. D)$\mu$= 2 and$\sigma$ = 1.5.

Accepted Answer

The mean is the number of questions on which the students agree, which is 7. The variance is the number of questions on which the students agree, minus the square of the mean, which is 1.5.

Question 13

For which of the following counts would a binomial probability model be reasonable?
A)the number of phone calls received in a one-hour period
B)the number of hearts in a hand of five cards dealt from a standard deck of 52 cards that has been thoroughly shuffled
C)the number of sevens in a randomly selected set of five digits from your table of random digits
D)All of the answer options are correct.

Accepted Answer

The answer of For which of the following counts would...

Question 14

Suppose we select an SRS of size n = 100 from a large population having proportion p of successes. Let X be the number of successes in the sample. For which value of p would it be safe to assume the sampling distribution of X is approximately Normal?
A)0.01
B)1/9
C)0.975
D)0.9999

Accepted Answer

The sampling distribution of $X$ can be approximated by a normal distribution if the sample size $n$ and population proportion $p$ satisfy the conditions $np \geq 10$ and $n(1-p) \geq 10$. For $n = 100$, option B) $1/9 \approx 0.111$ satisfies both conditions $100 \times \frac{1}{9} \geq 10$ and $100 \times \frac{8}{9} \geq 10$, making the normal approximation appropriate. Options A, C, and D do not satisfy these conditions as effectively due to very high or very low values of $p$, leading to either $np < 10$ or $n(1-p) < 10$.

Question 15

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The mean of X is:
A)0.4.
B)0.8.
C)1.2.
D)1.6.

Accepted Answer

The answer of At a large Midwestern college, 4% of...

Question 16

A small class has 10 students. Seven of the students are male and three are female. You write the name of each student on a small card. The cards are shuffled thoroughly, and you choose one at random, observe the name of the student, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe the name, and replace it in the set. This is done a total of five times. Let X be the number of cards observed in these five trials with a name corresponding to a male student. The random variable X has which of the following probability distributions?
A)the Normal distribution, with mean 3 and variance 1
B)the binomial distribution, with parameters n = 5 and p = 0.3
C)the binomial distribution, with parameters n = 5 and p = 0.7
D)None of the answer options is correct.

Accepted Answer

The probability of observing a male student in any one trial is 7/10. Since the trials are independent and identically distributed, X follows a binomial distribution with parameters n = 5 (number of trials) and p = 7/10 (probability of success in any one trial). Thus, the correct answer is (C).

Question 17

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability that 2 or fewer horses among the 10 horses admitted have been infected with West Nile virus?
A)0.8202
B)0.1937
C)0.3874
D)0.3487

Accepted Answer

This problem can be solved using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success on a single trial, and (n choose k) is the binomial coefficient. In this case, n = 10 (horses), p = 0.15 (probability of a horse having West Nile virus), and we're interested in the probability of k = 0, 1, or 2 horses having the virus. We calculate the probabilities for each of these cases and sum them up to find the total probability of 2 or fewer horses being infected. The correct answer, 0.8202, reflects the cumulative probability of these scenarios, indicating that there's an 82.02% chance that 2 or fewer out of the 10 horses admitted will have West Nile virus.

Question 18

If X has a binomial distribution with 20 trials and a mean of 6, then the success probability p is:
A)0.3.
B)0.5.
C)0.75.
D)This cannot be determined without taking a sample.

Accepted Answer

The mean of a binomial distribution is given by the formula μ = np, where n is the number of trials and p is the success probability. Given μ = 6 and n = 20, we can solve for p: 6 = 20p, which gives p = 0.3.

Question 19

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The probability that X is at least 1 is:
A)0.22.
B)0.44.
C)0.56.
D)0.77.

Accepted Answer

The probability of getting at least one Hispanic student in a random sample of 20 students from a college where 4% of the students are Hispanic is 0.56. This is because the probability of not getting any Hispanic students in the sample is (0.96)^20 = 0.44, and the probability of getting at least one Hispanic student is 1 - 0.44 = 0.56.

Question 20

Which of the following statements is not true about the binomial distribution?
A)A random variable having a binomial distribution is a finite count, and the minimum value is zero.
B)The smallest value can be zero or an integer above zero.
C)The overall experiment consists of independent and identical trials.
D)The mean does not have to be an integer, even though the variable is a count.

Accepted Answer

The binomial distribution always starts with the possibility of zero successes (e.g., zero heads in a series of coin flips), making statement B incorrect. The minimum value of a binomially distributed random variable is always zero, not an integer above zero.

Quiz 12: Binomial Distributions

Suppose X is a random variable with the binomial distribution with n = 4 and p = 1/4. The probability that X is greater than or equal to 1 is:

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability at least one of the 10 horses has been infected with West Nile virus?

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability a randomly selected horse among the 10 new admits has West Nile virus?

Suppose we toss a fair coin repeatedly. We continue to do this until a tail is observed. Let X be the number of tosses required. Then X has:

Opinion polls find that 20% of American adults claim they never have time to relax. Suppose you take a random sample of 200 American adults, and you count the number X in your sample that claim they never have time to relax. Using the Normal approximation, the probability that X is at least 50 is:

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The standard deviation of X is:

For which of the following counts would a binomial probability model be reasonable?

Suppose we select an SRS of size n = 100 from a large population having proportion p of successes. Let X be the number of successes in the sample. For which value of p would it be safe to assume the sampling distribution of X is approximately Normal?

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The mean of X is:

A local veterinary clinic typically sees 15% of its horses presenting with West Nile virus. If 10 horses are admitted during July, what is the probability that 2 or fewer horses among the 10 horses admitted have been infected with West Nile virus?

If X has a binomial distribution with 20 trials and a mean of 6, then the success probability p is:

At a large Midwestern college, 4% of the students are Hispanic. A random sample of 20 students from the college is selected. Let X denote the number of Hispanics among them. The probability that X is at least 1 is:

Which of the following statements is not true about the binomial distribution?