1) A machine dispenses natural spring water into bottles so that, on average, each bottle has 12 ounces.  In fact, the amount of water in a bottle is a random variable, X, that has a normal distribution with mean 12 ounces (oz.) and SD .2 oz.
a) If the bottle has too little water, the customer will complain.  What's the probability that the bottle has less than 11.85 ounces?  11 oz?
b) One gallon contains 128 ounces.  What's the mean and SD in terms of gallons?
c) Suppose you drink two bottles of water.   What's the probability that you just had more than 24 ounces of water?  More than 25?
d) In a six-pack, how many total ounces of water do you expect there to be?  What's the SD on this total?
e) In a six-pack, what do you expect the average number of ounces to be?  What's the SD of the average?
f) Find the probability that an average of 6 randomly selected bottles will be less than 11.85 ounces.

2) Another bet on Roullette is the "split".  You place your money on the line that separates two numbers, thus betting that either number will appear.  (For a picture of a roullette betting table, and some bets, see http://www.prtourism.com/roull.htm)    This bet pays 17 to 1, which means if you place a dollar on the bet, and one of those two numbers comes up, you win $17.  Otherwise, you lose your dollar.
a) Let X represent the amount you win on a single "split" bet.  Write the pdf of X.
b) Find the expected value and SD of X.
c) Suppose you play twice.  Let Y represent your total winnings after two plays.  Find the expected value and SD of Y.
d) Write the pdf of Y.
e) What's the probability that your total winnings will be a positive number after two split bets?
f) Suppose you play 50 times.  What's the (approximate) probability that your total winnings will be positive?
g) Sketch the approximate pdf of Z, where Z is your total winnings after 50 split bets.

3) The Central Limit Theorem (CLT) applies to sums of many random variables, and also those sums when multiplied by  constants or when constants are added to them.  This means that the CLT can be applied to binomial random variables.  Why? Recall that a binomial random variable, X, represents the number of successes in n independent trials.  We can think of each trial as producing either a 0 (failure) or a 1 (success.)  Then let Y1 represent the outcome of the first trial.  Y2 represents the outcome of the second, and so on.  If we do this, then X = Y1 + Y2 + ... + Yn.
a) Sketch the pdf of Y1
b) Find the expected value and SD of Y1
c) Use the result in (b) to find the expected value and SD of X = Y1 + Y2 +....+Yn.
d) Suppose we toss a fair coin 10 times. Let X represent the number of heads.  We can calculate the exact probability that X > 6 (P(X=7) + P(X = 8) + ... + P(X = 10)) using the formula I gave you in class. This turns out to be 0.05468.  But the CLT says that X is approximately a normal random variable.  Use this fact to find an approximate probability that X > 6.
e) Suppose we toss a fair coin 100 times.  Use the CLT to find the approximate probability that the number of heads will be between 55 and 65.

NOTE: The normal distribution provides a good approximation to the binomial distribution whenever np >= 10 AND n(1-p) >= 10.

4) 45% of Californians ate sushi last month.  Suppose we take a random sample (with replacement) of 1000 Californians.  Let X represent the number in our sample who ate sushi last month.
a) Show why X is a binomial random variable.
b) How many people in our sample should we expect to have eaten sushi last month?
c) Use the normal approximation to find the probability that the number of people in our sample who ate sushi last month will exceed 500.

Section 7.1, p. 301
1,5

Section 7.2, p. 311
1,2,3

Section 7.3. p. 323
1,3