STAT 310 DISCUSSION 8
TA: Yi Chai
Office: 1335N MSC
Email: [email protected]
Webpage: http://www.stat.wisc.edu/∼chaiyi
Office Hours: 11:00-12:00pm T and 1:00-2:00pm Th or by appointment
1. Confidence Intervals
100(1 −α)%-C.I. Margin of error Length of C.I.
mean of N(µ, σ2)
known σ2¯
Y±zα/2×σ
√nzα/2×σ
√n2zα/2×σ
√n
unknown σ2¯
Y±tα/2,n−1×s
√ntα/2,n−1×s
√n2tα/2,n−1×s
√n
Proportions ˆp±zα/2rˆp(1 −ˆp)
nzα/2rˆp(1 −ˆp)
n2zα/2rˆp(1 −ˆp)
n
2. Sample size calculation:
•Estimation of mean of a normal distribution N(µ, σ2) with known σ2.
The sample size nsuch that the margin of error for the 1 −αconfidence interval for µis no
greater than a prescribed value δis
n>σ2(Zα/2
δ)2
•Estimation of proportion. The required sample size is
n>1
4(Zα/2
δ)2
3. Examples
•Example 1: (6.3.10 from the textbook.) How many times must we toss a coin to ensure that a
0.95 confidence interval, for the probability of heads on a single toss, has length less than 0.1,
0.05 and 0.01, respectively?
•Example 2: (6.3.12 from the textbook.) Suppose that a measurement on a population can be
assumed to be distributed N(µ, 2) where µis unknown and that the size of the population is very
large. A researcher wants to determine a 0.95 confidence interval for µthat is no longer than 1.
What is the minimum sample size that will guarantee this?
•Example 3: (6.3.15 from the textbook.) Generate 103samples of size n= 5 from the Bernoulli(0.5)
distribution. For each of these samples, calculate (6.3.6)
"¯x−z(1+γ)/2r¯x(1 −¯x)
n,¯x+z(1+γ)/2r¯x(1 −¯x)
n#
with γ= 0.95 and record the proportion of intervals that contain the true value. What do you
notice? Repeat this simulation with n= 20. What do you notice?
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