Download Study Guide for Exam 3

Study Guide for Exam 3
Math 203, Spring 2012
To prepare for the third exam, you should review chapter 7 in the textbook, as well
as the Poisson handout. You should also review your homework problems, as many exam
problems will resemble homework you’ve done. As a general guide, I recommend reviewing
the following topics.
Chapter 7: Statistics
• Probability Distributions (§§7.1, 7.2)
– Given a description of an experiment, find the probability distribution as a chart
and as a histogram.
– Given a histogram or pie chart or frequency chart, answer questions about the
distribution.
• Measures of Central Tendency (§7.4)
– Given a set of data, find the mean.
µ=
x1 + x2 + · · · + xN
N
– Given a set of data, find the median and mode.
median = midmost data point
mode=most common data point
– Given a probability distribution, find the expected value E(X).
E(X) = µ = x1 p1 + x2 p2 + · · · + xN pN
• Measures of Dispersion (§7.5)
– Given a population of data x1 , . . . , xN , find the variance and standard deviation
Var =
(x1 − µ)2 + (x2 − µ)2 + · · · + (xN − µ)2
;
N
√
σ=
Var
– Given a sample x1 , . . . , xn from the population, find the variance and standard
deviation
Var =
√
(x1 − x)2 + (x2 − x)2 + · · · + (xn − x)2
;
σ = Var
n−1
(Recall that x is the mean of the sample.)
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Exam 3 Study Guide
Math 203, Spring 2012
– Given a probability distribution, find the variance and standard deviation
Var = (x1 )2 p1 + (x2 )2 p2 + · · · + (xn )2 pn − µ2 ;
√
σ=
Var
– Given two histograms, judge by inspection which has the greater standard deviation.
• The Binomial Distribution (§7.3)
– This is where you do a two-outcome experiment repeatedly and count the number
of successes.
n = number of times you repeat the experiment
p = probability of success during each trial
q = 1 − p = probability of failure
X = number of successes
– Important facts about the binomial distribution:
Pr(X = k)
E(X) = µ
Var
σ
=
=
=
=
C(n, k)pk q n−k
np
npq
√
npq
• The Poisson Distribution (handout)
– This may apply when you’re counting the number of “hits” per unit of space or
time.
– The number λ is the average number of hits per unit.
– Compute the Poisson distribution for a given λ
Pλ (X = k) =
– For the Poisson distribution, µ = λ and σ =
λk −λ
e
k!
√
λ.
– Use the Poisson distribution to answer applied questions, as on p. 515 of the
handout.
– Given a table of data, determine whether they are randomly distributed by comparing them with the Poisson distribution.
• The Normal Distribution (§7.6)
– Be able to find probabilities regarding the standard distribution Z,
such as Pr(Z ≤ 1) and Pr(Z ≥ 2.3) and Pr(1 ≤ Z ≤ 2).
– Especially be able to use the following:
Pr(−c ≤ Z ≤ +c) = 1 − 2 Pr(Z ≤ −c)
Exam 3 Study Guide
Page 3 of 3
Math 203, Spring 2012
– If a random variable X is normally distributed, use its mean µ and standard
deviation σ to translate numbers into z-scores:
The z-score of k is
k−µ
σ
– Use the normal distribution to approximate the binomial distribution.
(See below.)
– I will give you a copy of Appendix A to use.
• Normal Approximation to the Binomial Distribution (§7.7)
– The binomial distribution can be approximated by the normal distribution with
µ = np and σ =
√
npq.
This is especially good for answering questions with “≤” or “≥” in them, like
“What is the probability he gets at least 10 hits in 30 at-bats?”
– To use the normal distribution to approximate Pr(X ≥ k) or Pr(X ≤ k), you
must adjust your k by 0.5. For example, to approximate Pr(X ≥ 7) in a binomial
distribution, you must find Pr(X ≥ 6.5).
– An overall strategy:
√
1. Compute µ = np and σ = npq.
2. Adjust your boundary number by 0.5. Draw a histogram to see whether to
adjust it up or down.
3. Pretend the distribution is normal, and solve using the techniques of §7.6.