Confidence Intervals, Continued
Confidence Level for a Population Mean µ:
Choose an SRS of size n from a population having unknown mean µ and known standard deviation, 𝜎. A level
C confidence interval for is
𝑥̅ ± 𝑧 ∗
𝜎
√𝑛
Here z* is the value with area C between –z* and z* under the standard normal curve. This interval is exact
when the population distribution is normal and is approximately correct for large n in other cases.
How confidence intervals behave:
What we would like:
o High confidence
o Small margin of error
Remember: For confidence intervals for population mean, the margin of error = 𝑧 ∗
𝜎
√𝑛
How can we make the margin of error smaller?
1. Make z* smaller. Would have to accept lower confidence level. Don’t want that!!!
2. σ gets smaller.
Less variation in the population
3. n gets larger.
Just like it works for standard deviation in sampling distributions. The bigger
n is, the less variability (less error).
Example 10.6, p. 550
99% confidence Interval = (281.5, 331.1)
90% confidence interval = (290.5, 322.1)
Notice the difference. When we raised our
confidence level, it made the margin of error
larger.
So how do we get a high confidence level with a small margin of error?
PLAN AHEAD!!!!! We get to choose our sample size!!!!
Example 10.7, p. 551
Margin of error must be lesson than 5.
𝑚≤5
𝑧∗
𝜎
√𝑛
≤5
For a 95% confidence level, the critical value is z* = 1.96. In Example 10.5, they told us that σ = 0.43.
𝜎
𝑧∗
≤5
√𝑛
1.96
Multiply both sides by √𝑛, then
divide by 5
.43
√𝑛
≤5
1.96 ∗ .43
≤ √𝑛
5
16.856 ≤ √𝑛
Square both sides to get rid of square
root.
284.125 ≤ 𝑛
So take a sample of 285 video screens. (Always round up!!!!)
Sample Size for Desired Margin of Error
To determine the sample size n that will yield a confidence interval for a population mean with a specified
margin of error m, set the expression for the margin of error to be less than or equal to m and solve for n.
𝜎
𝑧∗
≤𝑚
√𝑛
Cautions with Confidence Intervals:
Any formula is correct only in specific circumstances (Look at p. 553 in your book)
Data must be from an SRS from the population
Formula not correct for more complex sampling designs (block, matched-pairs, etc.)
Fancy formulas can’t rescue badly produced data
Outliers can have a large effect on the confidence interval
Most importantly: The margin of error in a confidence interval covers only random sampling errors
(difficulties, such as undercoverage and noresponse in a sample survey can cause additional errors that
may be larger than the random sampling error)
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