In the table below the random variable X represents the number of people waiting in line at a fast food restaurant during the lunch hour.

x

0 1 2 3 4 5 Probability 0.0110.035 0.285 0.304 0.239 0.126

a) Verify that this is a discrete probability distribution. b) Draw a probability histogram. c) Compute and interpret the mean of the random variable X. d) Compute the standard deviation of the random variable X. e) What is the probability that there are two people waiting in line for lunch? f) What is the probability that there are more than three people waiting in line for lunch?

A candy factory has a 80% probability of making a good gumdrop. You
work in the quality control department and take a simple random sample
of 8 gumdrops.

What is the probability of 7 gumdrops in your sample being good?

What is the probability of at least 4 good gumdrops in the sample?

What is the probability of getting no more than 2 bad gumdrops in your
sample?

What are the mean and standard deviation of the number of good
gumdrops over many samples of size 8?

If you increase the sample to 16 gumdrops, what is the probability of 7 gumdrops in your sample being good?

At a certain university 12% of students are left hand users. Suppose that a random sample of 8 students were admitted in 2015 is selected.

(a) What is the probability that none of them will be a left hand user?

(b) What is the probability that 5 of them are left hand users?

(c) What is the probability that at least 7 of them are left hand user?

(d) What is the probability that at most 7 of them are left hand user?

(e) What is the probability that between 4 and 6 (inclusive) of them are left hand user?

(f) What is the expected value and standard deviation of number of users with left hand?

2.In a high school, 16% are Freshmen, 14% are Sophomores, 38% are Juniors, and 32% are Seniors. Suppose 15 students are randomly selected. Find the probability that

a) 4 are Freshmen, 5 are Sophomores, and the rest are neither Freshmen nor Sophomores (in any order)

b) either exactly one is Senior and all the others are not Senior (in any order) or exactly one Freshman, one Sophomore, one Junior and all the others are Seniors ( in any order)

Note: For each part, just give the formula as an answer; numerical answer is not required.

3. If seven fair dice are rolled, what is the probability that the number 2 and the number 6 will appear the same number of times?

(Note: “Same number of times” includes both 2 and 6 not appearing at all)

In families with four children, you're interested in the probabilities for the different possible numbers of girls in a family. Use theoretical probability (assume girls and boys are equally alike),compile a five -column table with the headings, O through 4, for the five possible numbers of girl children in a four child family. Then using G for girls and B for Boys, list under each heading the various birth order ways of achieving that number of girls in a family. Use table to calculate the following probabilities:

1. The probability of 1 girl.

2. The probability of 2 girls.

3. The probability of 4 girls.

4. The probability the third child born is a girl.

The probability of 4 girls

The issues surrounding the levels and structure of executive compensation have gained added prominence in the wake of the financial crisis that erupted in the fall of 2008. Based on the 2006 compensation data obtained from the Securities and Exchange Commission (SEC) website, it was determined that the mean and the standard error of compensation for the 582 highest paid CEOs in publicly traded U.S. companies are $12.01 million and $11.38 million, respectively. An analyst randomly chooses 31 CEO compensations for 2006 a. Is it necessary to apply the finite population correction factor? b. Is the sampling distribution of the sample mean approximately normally distributed? c. Calculate the expected value and the standard error of the sample mean. d. What is the probability that the sample mean is more than $17 million?

a. Is it necessary to apply the finite population correction factor?

b. Is the sampling distribution of the sample mean approximately normally distributed?

c. Calculate the expected value and the standard error of the sample mean.

d. What is the probability that the sample mean is more than $17 million?

Consider two $60,000 investments – call them Investment A and Investment B. Both investments will earn $5,000 with a probability of 0.5 and $1,000 with a probability of 0.5. Investment A will use 100% equity financing (issuing stocks). Investment B will get $30,000 through issuing stocks and $30,000 through issuing bonds. Investment B must pay 4% interest on the bonds.

a. Calculate the expected returns on equity (returns after interest payments divided by the amount of equity) for Investment A and Investment B. Express the returns as a percentage.

b. If the investments earned the lower amount ($1,000), what is the rate of return on equity for Investment A and Investment B? If the investments earned the higher amount ($5,000), what is the return on equity for Investment A and Investment B?

c. Using your answers from ‘a’ and ‘b’, what is the standard deviation of the rate of return on equity in each case? Which investment has the highest expected returns on equity? Which has the lowest risk? What explains the difference in risk between the two investments?

The Food and Drug Administration (FDA) is asked to approve a new drug. The new drug should contain less than 25mg of the active ingredient “toxin”, which is assumed to have dangerous side effects. The FDA would like to restrict the error of “approving the drug despite its too high content of toxin” to a maximum risk of 5% (α ≤ 0.05). Let Xi : “ The content of toxin in the i-th pill [in mg].” ∼ N(µ, σ2 ) ∼ N(µ, 4). A simple random sample of n = 50 pills ( Xi ∼ i.i.d.) will be used for the test.

7. Given a significance level of α = 5% what is the highest probability of making a type II error?

8. In the sample, x¯ = 24.6. Compute the p-value. What do you conclude? [Write down the probability that you computed.]

9. Has a type I error occurred? Explain your answer.

10. Has a type II error occurred? Explain your answer.

What is the region with the highest amount of carbon in soils