STAT 401 Week 4
Geoffrey Thompson
6/13/2013
1
Example Problems
1. There was, apparently, an election last year. There were a lot of polls and
a lot of statisticians had fun with them.
To cut a long story short, here is a simulation of election results based on
polls in the month leading up to the election: Election Simulation. There
are 20 realizations. The output X is the number of electoral votes won by
the GOP candidate.
(a) Plot a histogram and a QQ-plot. Does this distribution look normal?
(b) What is the mean? What is the standard deviation? Assuming
normality, what is P (X > 270)? How many of the 20 observations
are greater than 270?
(c) Suppose we know this is a normal distribution with σ = 23. Test the
hypothesis µ > 270.
(d) Again, suppose we know this is a normal distribution with σ = 23.
Construct a confidence interval for µ with α = 0.05.
(e) Let α = 0.05. A common power to use is 1 − β = 0.8. Suppose
we estimate σ = 23. We are interested in the hypothesis µ > 270.
We want to have a power of .05 when the sample mean is 240 and
µ0 = 270. What sample size do we need?
(f) Suppose we do not know σ. Test the hypothesis µ > 270.
(g) Construct a confidence interval for µ with α = 0.05.
1
(h) 20 is a rather small number of simulations to run. Here is a file
with 10,000: 10,000 Election Simulations. The central limit theorem
is in full force here when it comes to estimating the mean. Plot a
histogram and a QQ plot. Does this appear normal?
(i) Assuming normality, estimate P (X > 270) from this data set.
(j) One method to find P (X > 270) without making any assumptions
is to see what percentage of observations are greater than 270. Is
this estimate higher or lower than the one using the assumption of
normality?
2. There is a list of the top 100 chess players in July 2000. It is very exciting.
We are interested in their ratings. The rating is in the column labelled
“rating”.
(a) Plot a histogram of the ratings. Plot a QQ plot. What is the mean?
What is the standard deviation? Does this look normal?
(b) For two players, A and B, suppose P(A beats B) = Φ((RA −RB )/σR ),
where RA and RB are A’s rating and B’s rating, respectively, and
σR is some constant.
If we know Anand has a 62% chance of beating Karpov, what is σR ?
See the table of ratings.
(c) Ratings are not precise measurements, but rather estimates of skill
based on performance. Suppose σX = 24 is the standard deviation of
any individual’s rating. i.e., if repeated measurements of one player’s
rating were taken, it would be a random variable with some mean and
a standard deviation of 24. Create a confidence interval for Anand’s
rating with α = .05.
(d) Test whether Morozovich’s rating is over 2700 (using σ = 24).
(e) There are 99 players on the list. If 10 are chosen at random, what is
the probability that 2 have a rating under 2600?
2
References
• Polling Information
• FIDE Top Ratings List
• STAT 401 Page
2