6. Histograms:
a) How is the histogram related to the frequency plot?
i. When does a histogram look exactly like a frequency plot?
b) What visual information do you get from a histogram?
c) How can you tell what interval contains the most data points?
d) What is the relation between histograms and the cumulative distribution function?

e) ​The probability of winning at American roulette is 1/38 .
What is your expected number of wheel spins to get your first winner? ​

1. Calculate the mean, the variance, and the standard deviation of the following discrete probability distribution.

2. A marketing firm is considering making up to three new hires. Given its specific needs, the form feels that there is a 60% chance of hiring at least two candidates. There is only a 5% chance that it will not make any hires and a 10% chance that it will make all three hires.

*Find the expected value and standard deviation of the number of hires.

7. (Sec. 3.2) Two fair six-sided dice are tossed independently. Let M = the minimum of the two tosses. For example, M(2, 5) = 2, M(4, 4) = 4, etc.

(a) What is the PMF of M? [Hint: just work out each probability individually by counting the number of outcomes which result in a specific value for M, i.e. find p(1), then p(2), and so on up to p(6)].

(b) Determine the CDF of M. (

c) Graph the CDF of M.

Gravely Gears (GG) makes gears using an automated machine costs $12,000 and has a 25% probability of breaking irreparably at the end of each year (assuming it was working in the previous year). The machine has a maximum five-year life and will be disposed of with zero value at the end of five years. The machine produces $4,000 of cash flow at the end of each year and the discount rate is 8% per year. What is the expected number of years the machine will lastand what would the value of the machine be? What is the NPV of the machine?

Gravely Gears (GG) makes gears using an automated machine costs $12,000 and has a 25% probability of breaking irreparably at the end of each year (assuming it was working in the previous year). The machine has a maximum five-year life and will be disposed of with zero value at the end of five years. The machine produces $4,000 of cash flow at the end of each year and the discount rate is 8% per year. What is the expected number of years the machine will last and what would the value of the machine be? What is the NPV of the machine?

In upstate New York, milk tanker trucks follow a daily routine, stopping at the same dairy farms every day. Farm output varies due to weather, time of year, number of cows, and other factors. From years of recorded data, the mean amount of milk delivered for processing by milk tanker trucks is 7800 liters, with a standard deviation of 170 liters. Suppose 50 trucks are randomly selected. Find the probability that the mean amount of milk delivered by these trucks is more than 7760 liters. Justify your answer.

Faculty positions in a school district are classified as either teacher or administrator based on primary duties and responsibilities. The table below shows information on the number of national board-certified and not board-certified teachers and administrators.

Determine the probability, ?(board-certified or teacher), that a randomly chosen faculty member is either board-certified or a teacher. Please enter your answer as a decimal, precise to at least four decimal places.

?(board-certified or teacher) =

1. The calories in a bacon double cheeseburger at a certain restaurant are quoted as being 1200 calories. A study was performed to see if this value is accurate and the resulting data are given below. Is there sufficient evidence to state that the true average number of calories is something other than 1200 at a significance level of 0.05? What about a significance level of 0.01? Assume that the calories in a bacon double cheeseburger follow a normal probability distribution.

1302 1274 1125 1271 1217 1202 1184 1247 1100 1396 1352 1298 1295 1220 1402

According to Nielsen Media Research, the average number of hours of TV viewing by adults (18 and over) per week in the United States is 36.07 hours. Suppose the standard deviation is 9.7 hours and a random sample of 45 adults is taken.

a. What is the probability that the sample average is more than 38 hours?

b.  Suppose the population standard deviation is unknown. If 71% of all sample means are greater than 35 hours and the population mean is still 36.07 hours, what is the value of the population standard deviation?