Question

1. An experiment consists of drawing two marbles from a box containing red, yellow, and green marbles. One event in the sample space is RR. What are all of the events in the sample space? (Note: order matters) (3)

2. In a certain large population, 46% of households have a total annual income of over $70,000. A simple random sample is taken of four of these households. What is the probability that more than 2 of the households in the survey have an annual income of over $70,000?                                                                (8)

3. Which would you expect to have a higher standard deviation: data scores that are spread out or data scores that are close together?   
4. In a survey of U.S. households, 588 had home computers while 722 did not. Use this sample to estimate the probability of a household having a home computer.

1. Home Security Systems is studying the time utilization of its sales force. A random sample of 40 sales calls showed that the representatives spend an average of ẍ = 44 minutes on the road with a standard deviation of s = 6 minutes for each sales call.

1. Create a line showing the mean and values for 4 standard deviations above and 4 standard deviations below the mean.                                                                                                                                                (2)

2. Use Chebyshev’s Theorem to find the values between which we can expect at least 89% of the data to fall.                                                                                                                                                                       (2)

3. At least what percent of the data can we expect to fall between 38 and 50 minutes using the Empirical Rule?                                                                                                                                                        (2)

A psychologist studied the number of puzzles subjects were able to solve in a five-minute period while listening to soothing music. Let X be the number of puzzles completed successfully by a subject. The psychologist found that X had a distribution where the possible values of X are 1, 2, 3, 4 with the corresponding probabilities of 0.2, 0.4, 0.3, and 0.1. Use this information to answer questions 20-23 below.

2. Does the above data form a probability distribution? Create the distribution and explain why or why not.(5)

3. Using the above data, what is the probability that a randomly chosen subject completes at least three puzzles in the five-minute period while listening to soothing music?                                                                   (2)

4. Using the above data, the mean µ of X is what? Show work.                                                                             (3)
5. Using the above data (on previous page), the standard deviation σ of X is what? Show work.                      (4)

1. Why would we use the 10-90 percentile range?                                                                                                  +2

2. A data set has a range of 348. Use the range rule of thumb to determine the standard deviation.        +1

A batch of 26 light bulbs includes 5 that are defective. Two light bulbs are randomly selected. If the random variable, X, represents the number of defective light bulbs which can be selected, what values can X have?                

Answer 1

Note: Allowed to solve only one question per post.
But solved two questions in this case.

In a certain large population, 46% of households have a total annual income of over $70,000. A simple random sample is taken of four of these households. What is the probability that more than 2 of the households in the survey have an annual income of over $70,000?

This is a binomial experiment with n = 4, x = 2 and p = 0.46

We need to find P(X>2)

P(X>2) = 1 - P(X<=2)

P(X<=2) = P(X=0) + P(X=1) + P(X=2)

P(X=0)

P(X=1)

P(X=2)

P(X<=2) = P(X=0) + P(X=1) + P(X=2) = 0.0850 + 0.2897 + 0.3702 = 0.7450

P(X>2) = 1 - P(X<=2)= 1- 0.7450= 0.255

Which would you expect to have a higher standard deviation: data scores that are spread out or data scores that are close together?   

data scores that are spread out will have a higher standard deviation.

Intitutively we can understand standard deviation as the average distance of each point from the mean. If the points are spread out then the the distance from the mean will be higher and hence the average will higher , which turn will cause the standard deviation to be higher.