STAT 371 DISCUSSION 4
TA: Lane Burgette
Office: 1245F MSC, 1300 University Avenue
E-mail: [email protected]
URL: www.stat.wisc.edu/Ėburgette/371.html or naviagate from stat.wisc.edu
Office Hours: M 1:15-2:15; T 9:25-10:25
1 Sampling distributions
ā¢Normal Distribution
Be familiar with the normal table, and the different ways we can use it. In particular,
remember that it gives us P(Xā¤x), so you may have to do some subtraction to get the
desired probability. Also remember that you may have to look for a probability inside the
table to get a quantile. Also, be prepared to standardize, using Z= (XāĀµ)/Ļ.
ā¢Central Limit Theorem
For pretty much any distribution you are going to come across, ĀÆ
Ynwill be normally distributed
with the same mean as each Yiand variance Ļ2/n, for large n. A rule of thumb is that this
holds for nā„30, if the distribution is not very skewed.
If Yis normally distributed in the first place, then we have the same result, but it is exact,
and holds for any sample size.
ā¢Normal Approximation to the Binomial
If np and n(1 āp) are both 5 or bigger, then a binomial with parameters nand pcan be
approximated by a normal, with mean np and variance np(1āp). This can be improved using
the continuity correction, which can be seen by writing P(Xā¤x) = P(X < x + 1) = P(Xā¤
x+.5), for Xbinomial.
2 Exercises
ā¢Suppose that heights of students on this campus are normally distributed, with mean 62
inches, and standard deviation 3 inches.
What is the probability that a randomly selected studentās height exceeds 66 inches?
If I select 12 students, what is the probability that exactly 5 of them are 66 inches or taller?
With the same 12, what is the probability that the average is greater than 66 inches?
Now I want to sample more people. How many do I need to sample so that I can use the
normal approximation to the binomial? How does this compare to the rule of thumb for the
CLT?
Instead, I decide to sample 81 people. Find the probability that 18 or fewer are taller than
66 inches.
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