You are now prepared to calculate the probability of an event, distinguish between discrete and continuous random variables, and generate the probability distribution for both types of random variable

DATA FILE

ASSIGNMENT 3

1. Of six cars produced at a particular factory between 8 am and 10 am last Monday morning, test runs revealed three of them to be “lemons.” Nevertheless, three of the six cars were shipped to Dealer A and the other three to Dealer B. Dealer A received all three lemons. What is the probability of this event occurring if, in fact, the three cars shipped to Dealer A were selected at random from the six produced?

1. Carbon monoxide (CO) is an odorless, colorless, highly toxic gas which is produced by fires as well as by motor vehicles and appliances that use carbon-based fuels. A study on unintentional CO poisoning of Colorado residents for a six-year period identified 981 cases. Each case was classified as fatal or nonfatal and by source of exposure. The number of cases occurring in each of the categories is shown in the accompanying table. Assume that one of the 981 cases of unintentional CO poisoning is randomly selected.

1. 1. List all sample points for this experiment.

   2. What is the set of all sample points called?

   3. Let A be the event that the CO poisoning is caused by fire. Find P(A).

   4. Let B be the event that the CO poisoning is fatal. Find P(B).

   5. Let C be the event that the CO poisoning is caused by auto exhaust. Find P(C).

   6. Let D be the event that the CO poisoning is caused by auto exhaust and is fatal. Find P(D).

   7. Let E be the event that the CO poisoning is caused by fire but is nonfatal. Find P(E).

2. Security analysts are professionals who devote full-time efforts to evaluating the investment worth of a narrow list of stocks. The following variables are of interest to security analysts. Which are discrete and which are continuous random variables?

   1. The closing price of a particular stock on the New York Stock Exchange.

   2. The number of shares of a particular stock that are traded each business day.

   3. The quarterly earnings of a particular firm.

   4. The percentage change in yearly earning between 2015 and 2016 for a particular firm.

   5. The number of new products introduced per year by a firm.

   6. The time until a pharmaceutical company gains approval from the US Food and Drug Administration to market a new drug.

1. Explain why each of the following is or is not a valid probability distribution for a discrete random variable x:

a.

b.

c.

d.

SOURCE: MBS, Section 4.2, Problem 4.12, p. 167.

1. The random variable x has the following discrete probability distribution:

   x
   p(x)	0.1	0.2	0.4	0.2	0.1

   1. Find P(x ≤ 3).

   2. Find P (x < 3).

   3. Find P(x = 7).

   4. Find P(x ≥ 5).

   5. Find P(x > 2).

   6. Find P(3 ≤x ≤9).

1. Consider the probability distribution shown here.

   x	-4	-3	-2	-1	0	1	2	3	4
   p(x)	0.02	0.07	0.10	0.15	0.30	0.18	0.10	0.06	0.02

   1. Calculate μ, Ϭ2, and Ϭ.

   2. Graph p(x). Locate μ, μ - 2Ϭ, and μ + 2Ϭ on the graph.

   3. What is the probability that x is in the interval μ ± 2Ϭ?

1. Suppose x is a binomial random variable with n = 3 and p = .3.

   1. Calculate the value of p(x), x = 0, 1, 2, 3, using the formula for a binomial probability distribution.

   2. Using your answers in part a, give the probability distribution for x in tabular form.

1. The efficacy of insecticides is often measured by the dose necessary to kill a certain percentage of insects. Suppose a certain does of a new insecticide is supposed to kill 80% of the insects that receive it. To test the claim, 25 insects are expose to the insecticide.

   1. If the insecticide really kills 80% of the exposed insects, what is the probability that fewer than 15 die?

   2. If you observed such a result, what would you conclude about the new insecticide? Explain your logic.

1. In studying the dynamics of fish populations, knowing the length of a species at different ages is critical, especially for commercial fishermen. A study was published of the length distributions of sardines inhabiting Japanese waters. At two years of age, fish have a length distribution that is approximately normal with μ = 20.20 centimeters (cm) and Ϭ = 0.65 cm.

   1. Find the probability that a two-year-old sardine inhabiting Japanese waters is between 20 and 21 cm long.

   2. A sardine captured in Japanese waters has a length of 19.84 cm. Is this sardine likely to be 2 years old?

   3. Repeat part b for a sardine with a length of 22.01 cm.

1. Suppose x is a normally distributed random variable with μ = 50 and Ϭ = 3. Find a value of the random variable, call it x0, such that

   1. P(x ≤ x0) = 0.8413

   2. P(x > x0) = 0.25

   3. P(x > x0) = 0.95

   4. P(41 ≤ x < x0) = 0.8630

   5. 10% of the values of x are less than x0.

   6. 1% of the values of x are greater than x0.

2. Suppose a random sample of n measurements is selected from a population with mean μ = 100 and variance Ϭ2 = 100. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample mean x-bar.

   1. n = 4

   2. n = 25

   3. n = 100

   4. n = 50

   5. n = 500

   6. n = 1,000

1. A random sample of n = 68 observations is selected from a population with μ = 19.6 and Ϭ = 3.2. Approximate each of the following probabilities.

   1. P (x-bar ≤ 19.6)

   2. P(x-bar ≤ 19)

   3. P(x-bar ≥ 20.1)

   4. P (19.2 ≤ x-bar ≤ 20.6)

1. A particular manufacturing process requires steel rods that are at least 3 meters in length. The rods are purchased in lots of 50,000. To determine whether the lot meets the required quality standards, 100 rods are randomly sampled from each incoming lot and the mean length of rods in the sample is calculated. The quality manager has decided to accept lots whose sample mean is 3.005 meters or more. Assume that the standard deviation of the rod lengths in a lot is 0.03 meter.

   1. If in fact each lot has a mean length of 3 meters, what percentage of the lots received by the manufacturer will be returned to the vendor (i.e., the supplier)?

   2. If in fact all of the rods in all of the lots received by the manufacturer are between 2.999 and 3.004 meters in length, what percentage of the lots will be returned to the vendor?

1. To determine whether a metal lathe that produces machine bearings is properly adjusted, a random sample of 36 bearings is collected and the diameter of each is measured.

   1. If the standard deviation of the diameters of the bearings measured over a long period of time is 0.001 inch, what is the approximate probability that the mean diameter of the sample of 36 bearings will lie within 0.0001 inch of the population mean diameter of the bearings?

   2. If the population of diameters has an extremely skewed distribution, how will your approximation in part a be affected?