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6.5 Determining the sample size
Recall that a confidence interval is of the form
point estimator ± margin of error (ME)
[ point estimator – ME , point estimator + ME ]
ME
[
point estimator – ME
ME
]
|
point estimator
point estimator + ME
Example. If a confidence interval is [33.9, 35.1] or 34.5 ± 0.6, then ME = 0.6
Margin of Error
 Confidence interval for µ:
 Confidence interval for p:
√
√
PROBLEM: Determine the sample size needed to estimate the target parameter with a
prescribed margin of error.
Example. The manufacturer wishes to estimate the mean inflation pressure to within .025 pound of its
true value with a 99% confidence interval. The standard deviation of inflation pressure is about 0.1
(pound). What sample size should be used?
α/2 = 0.005, z α/2 = 2.576, σ ≈ 0.1, SE = 0.025 → n ≥ (2.5762 × 0.12)/ 0.0252 =106.19 → n = 107
Example. Suppose a candidate wants to estimate voter support p within 3% with 95% confidence.
How large sample does she need,
a) if she knows that p is about .6 ?
α/2 = 0.025, z α/2 = 1.960, p ≈ 0.6 → n ≥ (1.9602×0.6×0.4)/ 0.032 =1024.2 → n = 1025
b) if she has no idea about p?
α/2 = 0.025, z α/2 = 1.960, p ≈ 0.5 → n ≥ (1.9602×0.5×0.5)/ 0.032 =1067.2 → n = 1062
Exercise 1. A confidence interval for µ is (174.6, 211.2).
a. What is ̅ = ………? What is the margin of sampling error ME = ……….?
Exercise 2. A 98% confidence interval for p is (0.23, 0.37).
a. What is ̂ = ……… ? What is the margin of sampling error ME = ………?
b. What is the sample size n = …………….?
Exercise 3 [6.60, p. 331]. If you wish to estimate a population mean with a margin of error of ME = .3
using a 95% confidence interval, and you know from prior sampling that σ2 is approximately equal to
7.2, how many observations would have to be included in your sample?
Exercise 4 [6.62, p. 332] In each case, find the approximate sample size required to construct a 95%
confidence interval for p that has sampling error of ME = .08.
a. Assume p is near .2.
b. Assume you have no prior knowledge about p, but you wish to be certain that your sample is large
enough to achieve the specified accuracy for the estimate.