The S&P 500 index current level is 3,000. The dividend yield on the index is equal to the riskfree

rate of interest. Given a volatility of the index of 25%:

a) Compute the probability that the index value in 6 months is greater than 3,300.

b) Compute the probability that the index value in 6 months is less than 2700.

c) Compute the probability that the index value in 6 months is between 2700 and 3300.

The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If​ convenient, use technology to find the probability. For a sample of n equals 32​, find the probability of a sample mean being less than 12,751 or greater than 12,754 when mu equals 12,751 and sigma equals 1.1.

A worker-operated machine produces a defective item with probability 0.06 if the worker follows the machine's operating instructions exactly, and with probability 0.09 if he does not. If the worker follows the instructions 90% of the time, what proportion of all items produced by the machine will be defective?

If an item is known to be defective, what is the probability that the worker who made it followed the instructions? If necessary, round final answer to four decimal places.

The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 13.61.

1. The lambda of this distribution is
2. The probability that the percent is larger than 4.36 is P(x ≥ 4.36) =
3. The probability that the percent is less than 14.22 is P(x ≤ 14.22) =
4. The probability that the percent is between 10.61 and 14.61 is P(10.61 ≤ x ≤ 14.61) =

A given drug test has 5% false positives and 2% false negatives. The state has 10% of drug users.

a) If the person tests negative, what is the probability that the person is a drug user?

b) What is the probability of a non-drug user tests negative twice in a row with independent tests?

c) What is the probability of someone testing twice negative in a row is a non-drug user?

In a survey of 625 random students who took Probability and Statistics course, 400 pointed to Probability as the most enjoyable part of the course, the remaining 225 students chose Statistics as the most enjoyable part of the course. a) Find 90% confidence interval for the proportion of all students who preferred Probability. b) Find 95% confidence interval for the proportion of all students who preferred Statistics.

The labor force participation rate is approximately 63% in the US (i.e. approximately

63% of eligible adults actually work). Suppose 100 adults are randomly selected (assume

independence).

(a) Use the binomial distribution to find the probability that exactly 70 work.

(b) Use the normal approximation to find the probability that exactly 70 work.

(c) Use the normal approximation to find the probability that 70 or less work.

2. Suppose that on average, there are 15 companies making their initial public offering of stock (IPO) each month. Write down the corresponding Poisson formula for a)-c), then use R to get the final answers.

a) What is the probability of few than 3 IPOs in a month?

b) What is the probability of at least 15 IPOs in a month?

c) What is the probability of few than 30 IPOs in a two-month period?

The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 8.76.

The lambda of this distribution is

The probability that the percent is larger than 3.24 is P(x ≥ 3.24) =

The probability that the percent is less than 9.79 is P(x ≤ 9.79) =

The probability that the percent is between 5.76 and 11.76 is P(5.76 ≤ x ≤ 11.76) =