A baseball hitter hits a home run about once every 10 times at bat. We are interested in the number of hits before the first home run happens.

1. Give the distribution of X including parameters, if any. i.e., X ~ ________()

2. How many times does he bat on the average before the first home run?
3. Suppose we are interested in the number of hits before the third home run. Give the distribution of X including parameters, if any. i.e., X ~ ________()
   1. ​​​​​​​Show the following two using R:
4. What is the probability that the he has 4 hits before the first home run?
5. What is the probability that he has 7 hits before the third home run?

Compare samples of helium and neon gases at 25°C.

(a) Which gas has the highest average molecular speed? Explain your answer.

(b) Which gas has the highest average molecular kinetic energy? Explain your answer.

Suppose that a principal of a local high school tracks the number of minutes his students spend texting on a given school day. He finds that the distribution of minutes spent texting is roughly normal with a mean of 60 and a standard deviation of 20. Use this information to answer the following questions.

1. Based on the statistics, what is the probability of selecting at random a student who spends an extreme amount of time texting – either less than 10 minutes OR more than 110 minutes?

2. Based on the statistics, what is the probability of selecting at random (with replacement) two students who spent a below-average amount of time texting?

3. Based on the statistics, what is the probability of selecting at random (with replacement) two students who spent more than 75 minutes texting?

4. Based on the statistics, what is the percentile rank of a student who spent 100 minutes texting?

5. Based on the statistics, find the two numbers of minutes that define the middle 95% of students in the distribution. What is the value for the lower number that you found?

6. Based on the statistics, find the two numbers of minutes that define the middle 95% of students in the distribution. What is the value for the higher number that you found?

1. Many games require rolling 2 dice and adding the rolls together. Fill in the table below with the sum of the two die rolls. The first few cells have been completed as an example.

a. We assume die rolls are all equally likely. There are 36 possible outcomes (6x6) when we give them as ordered pairs like (2, 3), but when we look at adding them together, we get sums 2, 3, 4, 5, etc.

Complete this table with the sum of two dice, and the probability of each sum. (If you use decimals, use at least 3 digits, like 2.78%.)

b. Which number is the most likely? Which are the next most likely?

c. Now we’re going to compare the theoretical distribution (part a) with some empirical data.

• Roll 2 dice, and record the sum. Do this 30 times and put the results in the table above. (If you don’t have dice handy, use the Excel command =RANDBETWEEN(1,6) to roll a die.)
• Write a couple sentences about how your results compare with the theoretical distribution.
• What does the Law of Large Numbers say about what should happen to your results if you roll thousands and thousands of times?

2. We are going to roll a die with 20 sides, numbered 1 – 20. Each number on a die is assumed to be equally likely, but let’s mix things up a bit here.

Let’s say A = the number is 1 – 10, B = the number is 11 – 12, and C = the number is 13 – 20

a. Let’s say you roll the die once. Give the probability of each outcome A, B, and C.

(Make sure P(A) + P(B) + P(C) = 1.)

b. Suppose you roll the die two times. Now you have sequences like AA, AB, etc. Complete the table with all the possible sequences, and the probability of each sequence.

c. Make sure the probabilities add up to 1.

d. What is the probability that you get a two-roll sequence with no A’s in it?

An electron (mass m) is contained in a cubical box of widths L_x = L_y = L_z = L. It emits and absorbs light by making transitions among the lowest five energy levels. (a) How many different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energy levels? What multiple of h/8mL² gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and (g) third highest frequency?

A tank of water is emptied by the force of gravity through a syphon. The difference in water levels between the two tanks is 3 m and the highest point of the syphon is 2 m above the top surface level and the length of the pipe from inlet to the highest point is 2.5 m. The pipe is designed with a bore of 25 mm and the length 6 m. The pipe frictional coefficient is 0.007 and the inlet loss coefficient K is 0.7. Calculate the following:
4.1 the volume flow rate and,
4.2 the pressure at the highest point in the pipe

Question 2

Suppose the waiting time of a bus follows a uniform distribution on [0, 20]. (a) Find the probability that a passenger has to wait for at least 12 minutes. (b) Find the mean and interquantile range of the waiting time.

Question 3
Each year, a large warehouse uses thousands of fluorescent light bulbs that are burning 24 hours per day until they burn out and are replaced. The lifetime of the bulbs, X, is a normally distributed random variable with mean 620 hours and standard deviation 20 hours.

1. (a) If a light bulb is randomly selected, how likely its lifetime is less than 582 hours?

2. (b) The warehouse manager orders a shipment of 500 light bulbs each month. How

   many of the 500 bulbs are expected to have a lifetime that is less than 582 hours?

3. (c) The supplier of the light bulbs and the manager agree that any bulb whose lifetime

   is among the lowest 1% of all possible lifetimes will be replaced at no charge. What is the maximum lifetime a bulb can have and still be among the lowest 1% of all lifetimes?

Question 4
60% of students go to HKUST by bus. There are 10 students in the classroom. (a) What is the probability that exactly 5 of the students in the classroom go to

HKUST by bus?
(b) What is the mean number of students going to HKUST by bus?

Question 5
Peter is tossing an unfair coin that the probability of getting Head is 0.75. Let X be the random variable of number of trails that Peter get the first Head.
(a) Find the probability that the number of trails is five.
(b) Find the mean and standard deviation of random variable, X.

Question 6

Given that the population proportion is 0.6, a sample of size 1200 is drawn from the population.

1. (a) Find the mean and variance of the sampling distribution of sample proportion.

2. (b) Find the probability that the sample proportion is less than 0.58.

3. (c) If the probability that the sample proportion is greater than k is 0.6, find the value of k.

Consider the piston ring data in the following table. Assume that specifications are 74.00 ± 0.035 mm. Estimate the process capability (C_p and C_pk) using:

1. Sample Range Measurements
2. Sample Std Dev. Measurements

Convert the C_p found above into approximate dpm.

Inside Diameter Measurements (mm) for Automobile Piston Rings

The inside diameter of a randomly selected piston ring is a random variable with mean value 11 cm and standard deviation 0.06 cm.

(a) If

X

is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of

X

centered and what is the standard deviation of the

X

distribution? (Enter your standard deviation to five decimal places.)

(b) Answer the questions posed in part (a) for a sample size of n = 64 rings. (Enter your standard deviation to five decimal places.)

(c) For which of the two random samples, the one of part (a) or the one of part (b), is

X

more likely to be within 0.01 cm of 11 cm? Explain your reasoning.

X

is more likely to be within 0.01 cm of 11 cm in sample (b) because of the decreased variability with a larger sample size.

X

is more likely to be within 0.01 cm of 11 cm in sample (a) because of the increased variability with a smaller sample size.    

X

is more likely to be within 0.01 cm of 11 cm in sample (a) because of the decreased variability with a smaller sample size.

X

is more likely to be within 0.01 cm of 11 cm in sample (b) because of the increased variability with a larger sample size.