Question
1. Basketball. Suppose that against a certain opponent the number of points the MIT basketaball team scores is normally distributed with unknown mean ? and
1. Basketball. Suppose that against a certain opponent the number of points the MIT
basketaball team scores is normally distributed with unknown mean ? and unknown variance, ?2. Suppose that over the course of the last 10 games between the two teams MIT
scored the following points:
59, 62, 59, 74, 70, 61, 62, 66, 62, 75
(a) Compute a 95% t-confidence interval for ?. Does 95% confidence mean that the
probability ? is in the interval you just found is 95%?
(b) Now suppose that you learn that ?2 = 25. Compute a 95% z-confidence interval for
?. How does this compare to the interval in (a)?
(c) Let X be the number of points scored in a game. Suppose that your friend is a
confirmed Bayesian with a priori belief ? ? N(60, 16) and that X ? N(?, 25). He computes
a 95% probability interval for ?, given the data in part (a). How does this interval compare
to the intervals in (a) and (b)?
(d) Which of the three intervals constructed above do you prefer? Why?
2. The volume in a set of wine bottles is known to follow a N(, 25) distribution. You
take a sample of the bottles and measure their volumes. How many bottles do you have to
sample to have a 95% confidence interval for with width 1?
3. Suppose data x1, . . . , xn are i.i.d. and drawn from N(, ?2), where and ? are unknown.
Suppose a data set is taken and we have n = 49, sample mean x = 92 and sample standard
deviation s = 0.75.
Find a 90% confidence interval for .
4. You do poll to see what fraction p of the population supports candidate A over
candidate B.
(a) How many people do you need to poll to know p to within 1% with 95% confidence.
(b) Let p be the fraction of the population who prefer candidate A. If you poll 400 people,
how many have to prefer candidate A to so that the 90% confidence interval is entirely
above p = 0.5
11. Suppose we have 8 teams labeled T1, . . . , T8. Suppose they are ordered by placing
their names in a hat and drawing the names out one at a time.
(a) How many ways can it happen that all the odd numbered teams are in the odd
numbered slots and all the even numbered teams are in the even numbered slots?
(b) What is the probability of this happening?
12. (Taken from the book by Dekking et. al. problem 4.9) The space shuttle has 6 O-rings
(these were involved in the Challenger disaster). When launched at 81? F, each O-ring has
a probability of failure of 0.0137 (independent of whether other O-rings fail).
(a) What is the probability that during 23 launches no O-ring will fail, but that at least
one O-ring will fail during the 24th launch of a space shuttle?
(b) What is the probability that no O-ring fails during 24 launches
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