The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 71 and standard deviation 3.

a)If a specimen is acceptable only if its hardness is between 70 and 74, what is the probability that a randomly chosen specimen has an acceptable hardness? (Round your answer to four decimal places.)

b) If the acceptable range of hardness is (71 − c, 71 + c), for what value of c would 95% of all specimens have acceptable hardness? (Round your answer to two decimal places.)

c) If the acceptable range is as in part (a) and the hardness of each of ten randomly selected specimens is independently determined, what is the expected number of acceptable specimens among the ten? (Round your answer to two decimal places.)
d)What is the probability that at most eight of ten independently selected specimens have a hardness of less than 73.52? [Hint: Y = the number among the ten specimens with hardness less than 73.52 is a binomial variable; what is p?] (Round your answer to four decimal places.)

1. (5 pts) You are going to roll a pair of dice 108 times and record the sum of each roll. Before beginning, make a prediction about how you think the sums will be distributed. (Each sum will occur equally often, there will be more 12s than any other sum, there will be more 5s than any other sum, etc.) Record your prediction here:

2. (5 pts) Roll the dice 108 times and record the sum of each roll in the table provided below.

3. (22 pts) Find the experimental probability of rolling each sum. Fill out the following table:

4. (5 pts) Compare your outcomes to your prediction. Was your prediction correct? Why do you think this happened?

5. (5 pts) What number(s) occurred most often? Least often? Why do you think that is?

1. National Bank (see previous problem: pasted down below) is considering adding a second teller to the lunch-time situation to alleviate congestion. If the second teller is added, find the following:
   1. The average teller utilization
   2. The probability that there are 0 customers in the system
   3. The average number of customers in line
   4. The average time a customer waits before it seeing the teller
   5. The average time a customer spends in the service system
   6. If the tellers are paid $15/hour, is it worth adding the second teller (Hint: answer the question by only looking at the single hour by comparing 1 vs. 2 people)

To help answer the question, here is the "previous problem":

1. National Bank currently employs a single teller to assist customers over their lunch breaks. The typical arrival rate of customers is 11 people per hour where the teller can service people at a rate of 12 customers per hour. Assuming the standard assumptions of queuing models are met, find the following:
   1. The utilization of the teller
   2. The probability that there are 0 customers in the system, 8 customers in the system.
   3. The average number of customers in line
   4. The average time a customer waits before it seeing the teller
   5. The average time a customer spends in the service system

In a study on the physical activity of people, researchers measured overall physical activity as the total number of registered movements (counts) over a period of time and then computed the number of counts per minute (cpm) for each subject. The study revealed that the overall physical activity of obese people has a mean of μ=322cpm and a standard deviation σ=92cpm. In a random sample of 100 obese people, consider x ̅, the sample mean counts per minute.

Now suppose we are interested in the total activity (Σcpm) of a random sample of 25 obese people. Answer the following questions on a separate sheet of paper and upload to "Quiz Ch. 7 Free Response Answer":

a) Write a distribution statement that describes this new study of total activity (Σcpm).

b) Find the probability that the total cpm for 25 obese people is less than 7,500 cpm.

c) Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability in part b.

d) How likely is it that those 25 people will collectively register more than 9,000 cpm? Explain how you came to that conclusion.

6. There are 40 books on a bookshelf. Exactly 10 of them have a red cover. The remaining books have a white cover. You intend to choose a random sample of nine books, but you haven't decided whether to choose with replacement or without replacement?

a) If you choose books with our placement, would this procedure lead to independent trials or dependent trials?

b)if you choose books with replacement, would this procedure be consistent with a binomial experiment or not?

c) what is the size of the population? show your work

d) what is the size of the sample? show your work

e) if you ultimately conduct a binomial experiment, find the probability that the 5th book you select, happens to have a red cover? please show your work

f) if you ultimately conduct a binomial experiment, find the probability that you choose two or more red-covered books? please show your work

Finally, for a binomial experiment, please determine: show your work

1. average number of red books, per sample of nine books?
2. the variance of the number of red books, per sample of nine books?

Trucks are required to pass through a weighing station so that they can be checked for weight violations. Trucks arrive at the station at the rate of 35 an hour between 7:00 p.m. and 9:00 p.m. Currently two inspectors are on duty during those hours, each of whom can inspect 25 trucks an hour.
Use Table 1.

a. How many trucks would you expect to see at the weighing station, including those being inspected? (Round your answer to 3 decimal places.)

L_s             trucks

b.If a truck was just arriving at the station, about how many minutes could the driver expect to be at the station? (Round your answer to 2 decimal places.)

W_s             min.

c. What is the probability that both inspectors would be busy at the same time?(Round your answer to 4 decimal places.)

P_w           

d. How many minutes, on average, would a truck that is not immediately inspected have to wait? (Round your answer to the nearest whole number.)

W_a             min.

f. What is the maximum line length for a probability of .97? (Round up your answer to the next whole number.)

L _max