Question 1

A new drug is designed to reduce a person's blood pressure. Thirteen randomly selected hypertensive patients receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is untreated is 0.5. Then what is the probability of observing 11 or more blood pressure drops in a random sample of 13 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal places.

Accepted Answer

The answer of A new drug is designed to reduce...

Question 2

If x is a binomial random variable, calculate σ2 for n = 70 and p = 0.2.

Accepted Answer

The answer of If x is a binomial random variable,...

Question 3

If x is a binomial random variable, calculate μ for n = 70 and p = 0.4.

Accepted Answer

The mean (μ) of a binomial distribution is calculated using the formula μ = n * p. Here, n = 70 and p = 0.4, so μ = 70 * 0.4 = 28.

Question 4

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential election.

Accepted Answer

The answer of A recent study suggested that 70% of...

Question 5

According to a published study, 1 in every 4 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty-five men and asked each whether he has been involved in a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic accident? Round to the nearest whole number.

Accepted Answer

If 1 in 4 men has been involved in a minor traffic accident, then 3 in 4 men have not. Therefore, for 25 men, we expect 25 * (3/4) = 18.75, which rounds to 19 men who have never been involved in a minor traffic accident.

Question 6

A literature professor decides to give a 15-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than .10. What score should be set as the lowest passing grade?

Accepted Answer

The probability of guessing correctly on a true-false question is 1/2. The probability of guessing correctly on all 15 questions is (1/2)^15 = 1/32768. Therefore, the probability of passing a student who guesses on every question is less than .10 if the passing grade is set at 11 or higher.

Question 7

We believe that 82% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. If the true percentage is really 82%, find the probability of observing 38 or more students who consider statistics to be an exciting subject. Round to six decimal places.

Accepted Answer

The probability of observing 38 or more students who consider statistics to be an exciting subject is 0.004161. This can be calculated using the binomial distribution with n = 39 and p = 0.82.

Question 8

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 42% of all employees believe their company president possesses low ethical standards. Assume that responses were randomly and independently collected. A president of a local company that employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many of her company's employees believe that she possesses low ethical standards?

Accepted Answer

If 42% of the company's 1,000 employees believe the president possesses low ethical standards, then 0.42 * 1000 = 420 employees.

Question 9

We believe that 90% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 24 students from the population and observed fewer than five in our sample who consider statistics to be an exciting subject. Make an inference about the belief that 90% of the students consider statistics to be an exciting subject.

Accepted Answer

If 90% of students consider statistics exciting, we expect about 21.6 out of 24 to agree. Observing fewer than 5 suggests the 90% belief is likely too high.

Question 10

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo.

Accepted Answer

The answer of It a recent study of college students...

Question 11

If x is a binomial random variable, calculate σ for n = 100 and p = 0.3. Round to three decimal places when necessary.

Accepted Answer

The standard deviation of a binomial random variable is given by σ = √(np(1-p)). Substituting n = 100 and p = 0.3, we get σ = √(100 * 0.3 * 0.7) = 4.583 (rounded to three decimal places).

Question 12

An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 14 of these cars are independently sampled, what is the probability that more than half need new gas tanks?

Accepted Answer

The answer of An automobile manufacturer has determined that 30%...

Question 13

About 40% of the general population donate time and energy to community projects. Suppose 15 people have been randomly selected from a community and each asked whether he or she donates time and energy to community projects. Let x be the number who donate time and energy to community projects. Use a binomial probability table to find the probability that more than five of the 15 donate time and energy to community projects.

Accepted Answer

The answer of About 40% of the general population donate...

Question 14

The probability that an individual is left-handed is 0.16. In a class of 80 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth when necessary.

Accepted Answer

The mean is calculated as np = 80 * 0.16 = 12.8. The standard deviation is calculated as sqrt(np(1-p)) = sqrt(80 * 0.16 * 0.84) = 3.28.

Question 15

A local newspaper claims that 70% of the items advertised in its classifieds section are sold within 1 week of the first appearance of the ad. To check the validity of the claim, the newspaper randomly selected n = 25 advertisements from last year's classifieds and contacted the people who placed the ads. They found that 16 of the 25 items sold within a week. Based on the newspaper's claim, is it likely to observe $x \leq 16$ who sold their item within a week? Use a binomial probability table.

Accepted Answer

The answer of A local newspaper claims that 70% of...

Question 16

According to a recent study, 1 in every 6 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse.

Accepted Answer

The probability of a woman being a victim of domestic abuse is 1/6. The probability of a woman not being a victim of domestic abuse is 5/6. The probability of more than 22 of the women sampled not being the victim of domestic abuse is:P(X > 22) = 1 - P(X <= 22) = 1 - 0.811313 = 0.188687

Question 17

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 44% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards. Round to six decimal places.

Accepted Answer

Using the binomial probability formula, the probability that more than eight but fewer than 12 employees believe the president possesses low ethical standards is calculated as P(9 ≤ X ≤ 11) = P(X=9) + P(X=10) + P(X=11), which equals approximately 0.437608.

Question 18

According to a recent study, 1 in every 9 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that at least 2 of the women sampled have been the victim of domestic abuse. Round to six decimal places.

Accepted Answer

The probability of a woman being a victim is 1/9. Using the binomial distribution formula, calculate the probability of 0 or 1 victim and subtract from 1 to find the probability of at least 2 victims.

Question 19

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election.

Accepted Answer

The probability that more than 10 but fewer than 16 voters will vote is the sum of the probabilities for 11, 12, 13, 14, and 15 voters. Using a binomial probability table with n=20 and p=0.7, this sum is approximately 0.714.

Question 20

A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 20 adults over 50, what is the mean and standard deviation of the number who wear glasses? Round to the nearest hundredth when necessary.

Accepted Answer

The mean is calculated as np = 20 * 0.63 = 12.6. The standard deviation is calculated as sqrt(np(1-p)) = sqrt(20 * 0.63 * 0.37) ≈ 2.16.

Quiz 4: Discrete Random Variables