The life of a new gadget you invented to water a plant is normally distributed with a mean of 3 days with a standard deviation of 10 hours.   You took a sample of 16 gadgets.

A. What is the probability that the sample mean will be more than 3.2 days?

B. What is the probability that the sample mean would be between 2.8 and 3.2?

C. What is the probability that the sample mean would be either less than 2.75 or greater than 2.75?

The arrival rate of a customer at a service window of a cinema hall follows a probability distribution with a mean rate of 45 per hour. The service rate of the clerk follow a poison distribution with a mean of 60 per hour

1. Queue length
2. Queue
3. Traffic
4. Total time spent in the
5. Total time spent in the system
6. Find the probability of having 0 in the
7. Find the probability of having 4 in the system
8. The proportion of time the service facility is idle (in %age)

A process is in control and normally distributed with ? control chart limits of 45 and 15. The subgroup size is 4. Suppose the process variance suddenly triples while process mean remains unchanged. What is the probability that the first subsequent subgroup average will fall outside the control limits? What are the ? probability and ARL? Suppose the process variance suddenly triples while process mean shifts downward to 10. What are the β probability and ARL now?

The average battery life os an iPhone X is μ=9.52 hours with a standard deviation of σ =1.86 hours.

a) What is the probability that a single iPhone X lasts less than 9.25 hours?

b) What is the probability that a sample of 50 iPhone X phones lasts more than 10 hours?

c) What is the probability that a sample of 50 iPhone X phones lasts between 9.25 and 10 hours?

Assume that women's weights are normally distributed with a mean given by μ=143 lb and a standard deviation given by σ=29 lb.
(a)  If 1 woman is randomly selected, find the probability that her weight is above 177

(b)  If 6 women are randomly selected, find the probability that they have a mean weight above 177

(c)  If 78 women are randomly selected, find the probability that they have a mean weight above 177

A candy machine has 9 buttons of which one never works, two work one-half the time, and the rest work all the time. A coin is inserted and a button is pushed at random.

a) What is the probability that no candy is received?

b) If no candy is received, what is the probability that the button that never works was the one pushed?

c) If candy is received, what is the probability that one of the buttons that work one-half the time was the one pushed?

An urn contains 3 ​one-dollar bills, 1​ five-dollar bill and 1​ ten-dollar bill. A player draws bills one at a time without replacement from the urn until a​ ten-dollar bill is drawn. Then the game stops. All bills are kept by the player.​ Determine: ​(A)  The probability of winning ​$11 . ​(B)  The probability of winning all bills in the urn. ​(C)  The probability of the game stopping at the second draw.

A hospital administers a test to see if a patient has a certain disease. 2 % of the overall population has the disease. The test is 90% accurate.

(a)If a patient tests positive, what is the probability that they actually have the disease?

(b) If we instead perform two successive tests on each person, what is the probability that a person who tests positive both times actually has the disease? (Hint: drawing a probability tree might help)

9. For a standard normal distribution, what is the probability that Z is greater than -1.35? Round to four decimals and use leading zeros.

10. For a standard normal variable, what is the probability that Z is between -2.00 and -1.00? Round to four decimals and use leading zeros.

11. For a dataset that follows the standard normal distribution, what is the probability that Z is between 2.00 and 3.00? Round to four decimals and use leading zeros.