1.  Let X = the number of tosses of a coin until a Head first appears.  For example, you toss a coin and get (in this order) TTH, then X=3.
a) Find the pdf of X if the probability of a head on any given toss is p and assume p < 1.  (Hint, P(X=0)=0, P(X=1) = p).  This is called the geometric distribution.
b) What's the largest possible value of X?
c) What's the expected number of tosses before a H appears?  (Hint: Look up geometric sums in your calculus book.  If it only gives you the formula for finite sums (up to n), then take the limit as n goes to infinity.  (What is the limit of p^n as n goes to infinity if 0<p <1?)

2. For sufficiently large values of n, the binomial pdf can be approximated by a normal distribution with mean np and variance np(1-p).  For p = .5, an n of 100 is more than enough to make this a valid approximation.  Psally the Psychic claims to be able to guess whether a coin comes up heads or tails.  She does not claim perfect accuracy, but still claims to have a "sixth sense" that helps her do this.  If she is a fraud, then she just guesses like all of us do.
a) What's the approximate probability that Psally will get 60 or more coin flips correct in 100 tosses if she is just guessing?
b) Suppose its National Psychic Day and 10,000 people all participate in this experiment in which they guess coin flips in 100 flips of a fair coin.  About how many people do we expect to get 60 or more correct?

NOTE: THERE WERE CORRECTIONS MADE TO NUMBER 3 ON WED, JUNE 26.
3. In the last lab, you learned that the command (sample weight 6 t) would produce a random sample,WITH replacement, of size 6 from the weight variable in class. (Assuming you had downloaded this data.)  (If you omit the "t", then it samples without replacement.)
a) What's the average and SD of the weight data?
b) Assume for the moment that the weights follow a normal distribution? Imagine selecting a person at random. What would be the probability of being 12 pounds heavier OR MORE than the mean?  What is the actual probability for our data?
c) Suppose you select 5 people at random (with replacement).  What's the sampling distribution of their average WEIGHT?
d) What's the expected value for the average WEIGHT of these 5 people?
e) What's the Standard Error (SE) for the average?
f) What's the approximate probability that the average will be 12 pounds heavier OR MORE than the mean?
g) Select 5 people with replacement and calculate the average weight.  Repeat this 10 times.  What's the closest you get to the mean? What's the farthest?  What's the average of your averages?  what's the standard deviation of your averages?  How do these compare to the theoretical values you calculated above?

4. Select 10 people with replacement from the classdata and find their height.  Assuming heights follow a normal distribution with a standard deviation of 3 inches,  what's a 95% confidence interval for the mean height in the class?  Does it cover the true mean?  In this problem, we're treating the class as the population, and because we sample with replacement its as if this population were infinitely large with the probability distribution given exactly by the distribution of values in this class.  So the distribution of the population is not normal.  Explain why this might not matter.

5. Same question as 4, but now we make the more realistic assumption that the standard deviation is unknown.
 

6. Download the ARC data set rat.lsp from the data page.  The data represent a sample of rats, which we can assume were randomly sampled from a particular species of rat.
a) What's your estimate of the mean weight of  this species of rat?
b) What's your estimate of the SD of weights for this species of rat?
c) Calculate a 90% confidence interval for the mean weight.
d) Do the data look normal?  If not, how does this affect your confidence interval?

7. Download the blood pressure data from the data page.  These data represent systolic blood pressure of 15 patients before and after receiving medication designed to lower blood pressure.  Calculate the difference (after minus before) for all 15 patients.  (You do this by choosing "Add a variate" under the "Dataset" menu, and then typing "diff = after - before".  You don't have to call it "diff", of course.)
Is there evidence that the drug lowers blood pressure?  Or could these changes be explained just by chance variation?  State your assumptions, give evidence to support them (and if you can't, then state the possible effect on your conclusion should the assumptions not hold), and explain why you reached the conclusion you did.