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)