Question 1

A discrete random variable has a mean of 400 and a variance of 64.What is the standard deviation?
A)64
B)8
C)20
D)400

Accepted Answer

The standard deviation is the square root of the variance, so the standard deviation is √64 = 8.

Question 2

A company is considering producing some new Gameboy electronic games.Based on past records, management believes that there is a 70 percent chance that each of these will be successful and a 30 percent chance of failure.Market research may be used to revise these probabilities.In the past, the successful products were predicted to be successful based on market research 90 percent of the time.However, for products that failed, the market research predicted these would be successes 20 percent of the time.If market research is performed for a new product, what is the probability that the results indicate a successful market for the product and the product actually is successful?
A)0.90
B)0.54
C)0.60
D)0.63

Accepted Answer

Let's use Bayes' theorem to calculate the probability of a successful market research given a successful product.
P(Successful Market Research and Successful Product) = P(Successful Product) * P(Successful Market Research given Successful Product)
P(Successful Product) = 0.70
P(Successful Market Research given Successful Product) = 0.90
P(Failed Market Research given Failed Product) = 0.80
P(Failed Product) = 0.30

To calculate the probability of a successful market research overall, we need to consider the two ways in which a product can be successful: either the market research is successful and the product is successful, or the market research is failed and the product is successful.
P(Successful Market Research) = P(Successful Market Research and Successful Product) + P(Failed Market Research and Successful Product)
P(Failed Market Research and Successful Product) = P(Failed Market Research given Successful Product) * P(Successful Product) = 0.20 * 0.30 = 0.06
P(Successful Market Research) = 0.70 * 0.90 + 0.06 = 0.63

Therefore, the probability that the results indicate a successful market for the product and the product actually is successful is 0.63, which corresponds to answer choice D.

Question 3

A company is considering producing two new electronic games designed for the popular Gameboy toy.Based on market data, management believes that there is a 60 percent chance that a "cops and robbers" game will be successful and a 40 percent chance that a "let's play house" game will be successful.As these products are completely different, it may be assumed that the success of one is totally independent of the success of the other.If two products are introduced to the market, what is the probability that both are successful?
A)0.12
B)0.60
C)0.36
D)0.24

Accepted Answer

Since the success of one product is independent of the success of the other, the probability that both are successful is the product of their individual probabilities of success: 0.60 * 0.40 = 0.24.

Question 4

A dry cleaning business offers a pick-up and delivery service for a 10 percent surcharge.Management believes 60 percent of the existing customers will take advantage of this service.They are also considering offering customers the option of opening an account and receiving monthly bills.They believe 60 percent of customers (regardless of whether or not they use the pick-up service)will use the account service.If the two services are introduced to the market, what is the probability a customer uses neither of these services?
A)0.16
B)0.24
C)0.80
D)0.36

Accepted Answer

The probability of a customer using neither service is the complement of the probability of using either service. The probability of using either service is the sum of the probabilities of using each service minus the probability of using both services. The probability of using the pick-up service is 0.6, the probability of using the account service is 0.6, and the probability of using both services is 0.6 * 0.6 = 0.36. Therefore, the probability of using either service is 0.6 + 0.6 - 0.36 = 0.84. The probability of using neither service is 1 - 0.84 = 0.16.

Question 5

Properties of the normal distribution include
A)a continuous bell-shaped distribution.
B)a discrete probability distribution.
C)the number of trials is known and is either 1, 2, 3, 4, 5, etc.
D)the random variable can assume only a finite or limited set of values.

Accepted Answer

The normal distribution is a continuous probability distribution that is bell-shaped. It is not a discrete distribution, as it can take on any value within a certain range. The number of trials is not relevant to the normal distribution, and it can be used for both large and small sample sizes. The random variable also has an infinite number of possible values, not a finite or limited set.

Question 6

A dry cleaning business offers a pick-up and delivery service for a 10 percent surcharge.Management believes 60 percent of the existing customers will take advantage of this service.They are also considering offering customers the option of opening an account and receiving monthly bills.They believe 60 percent of customers (regardless of whether or not they use the pick-up service)will use the account service.If the two services are introduced to the market, what is the probability that a customer uses only one of these services?
A)0.40
B)0.60
C)0.48
D)0.24

Accepted Answer

Let P(P) denote the probability that a customer uses the pick-up service
Let P(A) denote the probability that a customer uses the account service

P(P) = 0.6 (given)
P(A) = 0.6 (given)

Let P(P and A) denote the probability that a customer uses both services
P(P and A) = P(P) * P(A) = 0.6 * 0.6 = 0.36

Now, we need to find the probability that a customer uses only one of these services. This can happen in two ways:

1) A customer uses only the pick-up service, not the account service.
In this case, the probability is P(P) - P(P and A) = 0.6 - 0.36 = 0.24

2) A customer uses only the account service, not the pick-up service.
In this case, the probability is P(A) - P(P and A) = 0.6 - 0.36 = 0.24

Adding these probabilities, we get:
P(using only one service) = 0.24 + 0.24 = 0.48

Therefore, the best choice is C) 0.48.

Question 7

A dry cleaning business offers a pick-up and delivery service for a 10 percent surcharge.Management believes 60 percent of customers will take advantage of this service.They are also considering offering customers the option of opening an account and receiving monthly bills.They believe 60 percent of their customers (regardless of whether or not they use the pick-up service)will use the account service.If the two services are introduced to the market, what is the probability a customer uses both services?
A)0.12
B)0.60
C)0.36
D)0.24

Accepted Answer

Let A be the event that a customer uses the pick-up and delivery service and B be the event that a customer opens an account.

We are given:
P(A) = 0.6 (60% of customers will use the pick-up service)
P(B) = 0.6 (60% of customers will open an account)
P(A|B) = P(A intersect B)/P(B)

We need to find P(A intersect B), the probability that a customer uses both services.

To do this, we can use the formula:
P(A intersect B) = P(B) * P(A|B) 
P(A intersect B) = 0.6 * P(A|B)

We don't have enough information to find P(A|B) directly, but we can use the fact that 10% of customers will be charged a surcharge for the pick-up service. This means that if a customer opens an account, they will also be charged the surcharge if they use the pick-up service.

So, if B occurs (a customer opens an account), then the probability that a customer will use the pick-up service and be charged the surcharge is: 
P(A|B) = 0.6 + 0.4*0.1 = 0.64

Now we can find: 
P(A intersect B) = 0.6 * 0.64 = 0.384

Therefore, the probability that a customer uses both services is 0.384 or 38.4%, which is closest to 0.36 (choice C).

Question 8

The number of phone calls coming into a switchboard in the next five minutes will either be 0, 1, 2, 3, 4, 5, or 6.The probabilities are the same for each of these (1/7).If X is the number of calls arriving in a five-minute time period, what is the mean of X?
A)2
B)3
C)4
D)5

Accepted Answer

To find the mean, we need to multiply each possible value of X by its probability of occurring, and then add up those products.

So, the mean of X can be calculated as follows:

E(X) = (0)(1/7) + (1)(1/7) + (2)(1/7) + (3)(1/7) + (4)(1/7) + (5)(1/7) + (6)(1/7)

E(X) = 3

Therefore, the mean of X is 3, which is option B

Question 9

The number of phone calls coming into a switchboard in the next five minutes will either be 0, 1, or 2.The probabilities are the same for each of these (1/3).If X is the number of calls arriving in a five-minute time period, what is the mean of X?
A)1/3
B)2/3
C)1
D)4/3

Accepted Answer

The mean (or expected value) of X is given by E(X) = 0*(1/3) + 1*(1/3) + 2*(1/3) = 1.

Question 10

The expected value of a probability distribution is
A)the measure of the spread of the distribution.
B)the variance of the distribution.
C)the average value of the distribution.
D)the probability density function.

Accepted Answer

The expected value of a probability distribution is the theoretical average value that the random variable takes. It represents the center of mass of the distribution and is calculated as the weighted average of all possible values of the random variable, where the weights are given by the probabilities associated with each value. Therefore, the correct option is (C) the average value of the distribution.

Question 11

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that between 2 and 5 (inclusive)customers are willing to switch companies?
A)0.1369
B)0.1746
C)0.0377
D)0.7350

Accepted Answer

The answer of Historical data indicates that only 20% of...

Question 12

Which of the following is not true for discrete random variables?
A)The expected value is the weighted average of the values.
B)They can assume only a countable number of values.
C)The probability of each value of the random variable must be 0.
D)The probability values always sum up to 1.

Accepted Answer

The probability of each value of the random variable must not necessarily be 0, but it should always be between 0 and 1, inclusive.

Question 13

Which distribution is helpful in testing hypotheses about variances?
A)binomial distribution
B)distribution
C)normal distribution
D)Poisson distribution

Accepted Answer

The chi-square distribution is specifically used in hypothesis testing related to variances. It helps in assessing whether a sample variance significantly differs from a hypothesized population variance.

Question 14

A company is considering producing some new Gameboy electronic games.Based on past records, management believes that there is a 70 percent chance that each of these will be successful, and a 30 percent chance of failure.Market research may be used to revise these probabilities.In the past, the successful products were predicted to be successful based on market research 90 percent of the time.However, for products that failed, the market research predicted these would be successes 20 percent of the time.If market research is performed for a new product, what is the probability that the product will be successful if the market research indicates a success?
A)0.10
B)0.90
C)0.91
D)0.63

Accepted Answer

Using Bayes' theorem, we can calculate the updated probability:
P(success | market research indicates success) = P(market research indicates success | success) x P(success) / P(market research indicates success)
P(market research indicates success | success) = 0.9 (given)
P(success) = 0.7 (given)
P(market research indicates success) = P(market research indicates success | success) x P(success) + P(market research indicates success | failure) x P(failure)
P(market research indicates success | failure) = 0.2 (given), P(failure) = 0.3 (given)
P(market research indicates success) = 0.9 x 0.7 + 0.2 x 0.3 = 0.67
Putting all the values into the formula:
P(success | market research indicates success) = 0.9 x 0.7 / 0.67 = 0.91
Therefore, the probability that the product will be successful if the market research indicates a success is 0.91.

Question 15

The number of cell phone minutes used by high school seniors follows a normal distribution with a mean of 500 and a standard deviation of 50.What is the probability that a student uses fewer than 600 minutes?
A)0
B)0.023
C)0.841
D)0.977

Accepted Answer

To find the probability that a student uses fewer than 600 minutes, we calculate the z-score: z = (600 - 500) / 50 = 2. Using the standard normal distribution table, the probability of z being less than 2 is approximately 0.977.

Question 16

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that exactly 2 customers would be willing to switch their cable?
A)0.1
B)0.04
C)0.137
D)0.206

Accepted Answer

We have n = 20, the probability of success (finding a customer willing to switch) is p = 0.20, and we want to find the probability of exactly 2 successes, so we use the binomial probability formula:

P(X=2) = (20 choose 2) * (0.20)^2 * (0.80)^18
         = 190 * 0.04 * 0.006
         = 0.1368

Therefore, the answer is C, 0.137.

Question 17

The number of cell phone minutes used by high school seniors follows a normal distribution with a mean of 500 and a standard deviation of 50.What is the probability that a student uses fewer than 400 minutes?
A)0
B)0.023
C)0.159
D)0.977

Accepted Answer

We need to find the probability of using fewer than 400 minutes. We can standardize this value using z-score formula.

z = (x - mean) / standard deviation
z = (400 - 500) / 50
z = -2

From the standard normal distribution table, the probability of getting a z-value of -2 or less is 0.023. Therefore, the probability that a student uses fewer than 400 minutes is 0.023.

Question 18

Which of the following is not true about continuous random variables?
A)They have an infinite set of values.
B)The area under each of the curves represents probabilities.
C)The entire area under each of the curves equals 1.
D)They can only be integer values.

Accepted Answer

Continuous random variables can take on any value within a range or interval, including non-integer values.

Question 19

A company is considering producing two new electronic games designed for the popular Gameboy toy.Based on market data, management believes that there is a 60 percent chance that a "cops and robbers" game will be successful and a 40 percent chance that a "let's play house" game will be successful.As these products are completely different, it may be assumed that the success of one is totally independent of the success of the other.If two products are introduced to the market, what is the probability that both are failures?
A)0.16
B)0.24
C)0.80
D)0.36

Accepted Answer

The probability that both games fail is the product of their individual failure probabilities: (1 - 0.60) * (1 - 0.40) = 0.40 * 0.60 = 0.24.

Question 20

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that no more than 3 customers would be willing to switch their cable?
A)0.85
B)0.15
C)0.20
D)0.411

Accepted Answer

The probability of success (a cable customer switching) is 20% or 0.2, and the probability of failure (a cable customer not switching) is 80% or 0.8. Therefore, using the binomial probability formula: P(X ≤ 3) = Σ(20 choose k) (0.2)^k (0.8)^(20-k) for k = 0, 1, 2, 3. Calculating this sum gives P(X ≤ 3) = 0.4108 or approximately 0.411. Therefore, the probability that no more than 3 customers would be willing to switch their cable is 0.411.

Quiz 2: Probability Concepts and Applications

A discrete random variable has a mean of 400 and a variance of 64.What is the standard deviation?

Properties of the normal distribution include

The number of phone calls coming into a switchboard in the next five minutes will either be 0, 1, 2, 3, 4, 5, or 6.The probabilities are the same for each of these (1/7) .If X is the number of calls arriving in a five-minute time period, what is the mean of X?

The number of phone calls coming into a switchboard in the next five minutes will either be 0, 1, or 2.The probabilities are the same for each of these (1/3) .If X is the number of calls arriving in a five-minute time period, what is the mean of X?

The expected value of a probability distribution is

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that between 2 and 5 (inclusive) customers are willing to switch companies?

Which of the following is not true for discrete random variables?

Which distribution is helpful in testing hypotheses about variances?

The number of cell phone minutes used by high school seniors follows a normal distribution with a mean of 500 and a standard deviation of 50.What is the probability that a student uses fewer than 600 minutes?

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that exactly 2 customers would be willing to switch their cable?

The number of cell phone minutes used by high school seniors follows a normal distribution with a mean of 500 and a standard deviation of 50.What is the probability that a student uses fewer than 400 minutes?

Which of the following is not true about continuous random variables?

Historical data indicates that only 20% of cable customers are willing to switch companies.If a binomial process is assumed, then in a sample of 20 cable customers, what is the probability that no more than 3 customers would be willing to switch their cable?