Question 1

A study conducted by a local university found that 25 percent of college freshmen support increased spending on environmental issues.If 6 college freshmen are randomly selected,find the probability that only 1 supports increased spending on environmental issues.

Accepted Answer

The answer of A study conducted by a local university...

Question 2

In the most recent election,19 percent of all eligible college students voted.If a random sample of 20 students were surveyed,find the probability that none of the students voted.

Accepted Answer

The probability of one student not voting is 0.81 (100%-19%). The probability of none of the 20 students to vote is (0.81)^20 = 0.0148. Therefore, the answer is C.

Question 3

The manager of the local grocery store has determined that,on average,4 customers use the service desk every half-hour.Assume that the number of customers using the service desk has a Poisson distribution.What is the probability that during a randomly selected half-hour period,no more than 2 customers use the service desk?

Accepted Answer

The answer of The manager of the local grocery store...

Question 4

The manager of the local grocery store has determined that,on average,4 customers use the service desk every half-hour.Assume that the number of customers using the service desk has a Poisson distribution.What is the probability that during a randomly selected half-hour period,exactly 2 customers use the service desk?

Accepted Answer

The answer of The manager of the local grocery store...

Question 5

In a study conducted for the state Department of Education,30 percent of the teachers who left teaching did so because they were laid off.Assume that we randomly select 10 teachers who have recently left their profession.Find the probability that exactly 4 of them were laid off.

Accepted Answer

The answer of In a study conducted for the state...

Question 6

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 100 calculators are selected at random,what is the expected number of defectives?

Accepted Answer

The expected number of defectives can be calculated by multiplying the total number of calculators with the probability of defectives. Therefore,
Expected number of defectives = 100 x 0.1 = 10. Thus, the best choice is C, which represents the expected number of defectives.

Question 7

According to a survey of adults,64 percent have money in a regular savings account.If we were to survey 50 randomly selected adults,find the mean number of adults who would have regular savings accounts.

Accepted Answer

The mean number of adults with savings accounts is calculated by multiplying the total number of adults surveyed (50) by the percentage with savings accounts (64%), which equals 32.

Question 8

A multiple-choice test has 30 questions and each one has five possible answers,of which one is correct.If all answers were guesses,find the probability of getting exactly four correct answers.

Accepted Answer

The probability of guessing a correct answer for each question is 1/5, and the probability of guessing an incorrect answer is 4/5.

Using the binomial probability formula, we can calculate the probability of getting exactly 4 correct answers out of 30:

P(X = 4) = (30 choose 4) (1/5)^4 (4/5)^26

Using a calculator, we can simplify this to approximately 0.1325. Therefore, the best choice is B, with a probability of 0.1325.

Question 9

Of all individual tax returns,37 percent include errors made by the taxpayer.If IRS examiners are assigned randomly selected returns in batches of 12,find the mean and standard deviation for the number of erroneous returns per batch.

Accepted Answer

The number of erroneous returns in a batch of 12 follows a binomial distribution with parameters n=12 and p=0.37. 
The mean of a binomial distribution is μ = np, so μ = 12 x 0.37 = 4.44. 
The standard deviation of a binomial distribution is σ = sqrt(np(1-p)), so σ = sqrt(12 x 0.37 x 0.63) = 1.67. 
Therefore, the best choice is B, with μ = 4.44 and σ = 1.67.

Question 10

A manufacturer tested a sample of semiconductor chips and found that 35 were defective and 190 were good.If additional tests are to be conducted with random samples of 160 semiconductor chips,find the mean for the number of defects in these groups of 160 (rounded to the nearest whole number).

Accepted Answer

We can use the formula for the mean of a binomial distribution: mean = np, where n is the sample size and p is the probability of success. In this case, the probability of choosing a defective chip is 35/(35+190) = 0.1556. Therefore, the mean for the number of defects in a sample of 160 is:

mean = 160 x 0.1556 ≈ 24.89 ≈ 25 (rounded to the nearest whole number)

Question 11

A study conducted by a local university found that 25 percent of college freshmen support increased spending on environmental issues.If 6 college freshmen are randomly selected,find the probability that fewer than 4 support increased spending on environmental issues.

Accepted Answer

The answer of A study conducted by a local university...

Question 12

In the most recent election,19 percent of all eligible college students voted.If a random sample of 20 students were surveyed,find the probability that exactly half voted in the election.

Accepted Answer

The probability of exactly half of the 20 randomly sampled students voting in the election can be calculated using the binomial probability formula: 
P(X=10) = (n choose x) * p^x * q^(n-x)
where n = 20, x = 10, p = 0.19 (probability of success) and q = 0.81 (probability of failure). 
Plugging in these values, we get: 
P(X=10) = (20 choose 10) * 0.19^10 * 0.81^10
= 0.0014
Therefore, the answer is B.

Question 13

A fair die is rolled 36 times.What is the standard deviation of the even number (2,4,or 6)outcomes?

Accepted Answer

The probability of rolling an even number on a fair die is 1/2. Therefore, the expected value of the number of even outcomes in 36 rolls is (1/2) * 36 = 18.

To calculate the standard deviation of the even outcomes, we need to use the formula:

SD = sqrt [ Σ(x - μ)^2 / N ]

where: 
- x is the number of even outcomes (2, 4, or 6) 
- μ is the mean of the even outcomes, which is 18 
- N is the number of rolls, which is 36

Let's fill in the formula:

SD = sqrt [ ( (2-18)^2 + (4-18)^2 + (6-18)^2 ) / 36 ]

= sqrt [ ( 16 + 196 + 256 ) / 36 ]

= sqrt [ 24 ]

= 4.899

Rounding this to the nearest whole number gives us a standard deviation of 5. Therefore, the answer is D.

Question 14

The probability that a given computer chip will fail is 0.02.Find the probability that of 5 delivered chips,exactly 2 will fail.

Accepted Answer

The answer of The probability that a given computer chip...

Question 15

A study conducted by a local university found that 25 percent of college freshmen support increased spending on environmental issues.If 6 college freshmen are randomly selected,find the probability that exactly 3 support increased spending on environmental issues.

Accepted Answer

This problem can be modeled using the binomial distribution since each college freshman can be considered a Bernoulli trial with a probability of success (supporting increased spending on environmental issues) of 0.25. The probability of getting exactly 3 successes in 6 trials can be calculated using the binomial probability formula: P(X=3) = (6 choose 3) * 0.25^3 * 0.75^3. Plugging this into a calculator or using a binomial probability table yields a probability of 0.1318 or approximately 13.18%. Therefore, the best choice is B.

Question 16

It is known that 15 percent of all college freshmen at Ivey University leave after their first year.Calculate the interval that contains 99.73 percent of the students who remain if we are looking at a random sample of 300 Ivey University freshmen.

Accepted Answer

The mean number of students who remain is $300 \times 0.85 = 255$. The standard deviation is $\sqrt{300 \times 0.15 \times 0.85} \approx 5.64$. Using the 68-95-99.7 rule, the interval is $255 \pm 3 \times 5.64$, which is approximately $[236.45, 273.55]$.

Question 17

A fair die is rolled 10 times.What is the average number of even number outcomes?

Accepted Answer

Binomial mean = μ = np = 10(.5) = 5

Question 18

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 10 calculators are selected at random,what is the probability that 3 or more of the calculators will be defective?

Accepted Answer

The answer of The J.O.Supplies Company buys calculators from a...

Question 19

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 3 calculators are selected at random,what is the probability that one of the calculators will be defective?

Accepted Answer

To solve the problem, we can use the binomial distribution formula:

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

where:
- P(X = k) is the probability of getting exactly k defective calculators
- nCk is the number of ways to choose k items out of n
- p is the probability of getting a defective calculator
- (1-p) is the probability of getting a non-defective calculator
- n is the total number of calculators selected (in this case, n=3)

To find the probability of getting one defective calculator, we need to calculate the probability of getting one defective calculator and two non-defective calculators, and then multiply by the number of ways we can choose which calculator is defective.

P(one defective calculator) = 3C1 * 0.1^1 * 0.9^2 = 0.243

Therefore, the answer is C) 0.2430.

Question 20

A company manufactures an appliance,gives a warranty and 95 percent of its appliances do not require repair before the warranty expires.If an organization buys 10 of these appliances,calculate an interval that contains 95.44 percent of all the appliances that won't require repair.

Accepted Answer

The number of appliances that won't require repair follows a binomial distribution with n=10 and p=0.95. Using the normal approximation, the mean is 9.5 and the standard deviation is approximately 0.69. For 95.44% (which corresponds to approximately 2 standard deviations), the interval is [8.12, 10.88].

Quiz 5: Discrete Random Variables