MATH 2400
Ch. 14 Notes
Statistical Estimation – an Example
A National Health and Nutrition Examination Survey (NHANES) report
gives data for 654 women age 20 to 29 years. The mean BMI of these
654 women was x̄ = 26.8. On the basis of this sample, we want to
estimate the mean BMI μ in the population of all 20.6 million women in
this age group.
We will treat the samples as an SRS from a Normal population with
standard deviation σ=7.5.
Statistical Estimation – In a Nutshell
Statistical Estimation – In a Nutshell
So, for this particular sample, this interval is
x̄ - 0.586 = 26.8 – 0.586 = 26.214
to
x̄ + 0.586 = 26.8 + 0.586 = 27.386
4. Because we got the interval 26.214 to 27.386 from a method that
captures the population mean for 95% of all possible samples, we say
that we are 95% confident that the mean BMI μ of all young women is
some value in that interval, no lower than 26.2 and no higher than
27.4.
Confidence Intervals
The 95% confidence interval for the mean BMI of young women, based
on the NHANES sample, is x̄ ± 0.586. Once we have the sample results
in hand, we know that for the sample x̄ = 26.8, so our confidence
interval is 26.8 ± 0.286. Most confidence intervals have this form.
estimate ± margin of error
A level C confidence interval for a parameter has two parts
1) An interval calculated from the data, usually in the form above, and
2) A confidence level C, which gives the probability that the interval
will capture the true parameter value in repeated samples. That is,
the confidence level is the success rate for the method.
Confidence levels can be chosen, but are usually 90% or higher because
we want to be quite sure of our conclusions. 95% is the most common.
Example 2 (exercise 14.4)
A Gallup Poll in November 2010 found that 54% of the people in the
sample said they want to lose weight. Gallup announced, “For results
based on the total sample of national adults, one can say with 95%
confidence that the maximum margin of sampling error is ±4
percentage points.”
What is the 95% confidence interval for the percent of all adults who
want to lose weight?
Typical Confidence Intervals
These are typical confidence
intervals and their corresponding
z-scores.
Conf. Interval:
Conf. Int. for Mean of Normal Pop.
Example 3
State: Does the expectation of good weather influence generosity? The following
represents the tips given by 20 patrons (in percentages) at a restaurant. The bill noted that
the next day’s weather would be good. There are also 2 more samples, one said the next
day’s weather would not be good, and the last had no message on the bill.
Plan: We will estimate the mean percentage tip μ for all patrons of this restaurant when
they receive a message on their bill indicating that the next day’s weather will be good by
giving a 95% confidence interval.
Solve: The mean percentage tip of the sample is x̄ = 22.21. Suppose that from past
experience with patrons of this restaurant we know that the standard deviation of
percentage tip is σ=2. For 95% confidence, the critical value is x = 1.960. A 95% confidence
interval for μ is
How Do We Know σ?
Chapter 18…coming later!
Example 4 (exercise 14.7)
The Goal…
High confidence and low margin of error.
BUT…
The margin of error gets smaller when…
• z gets smaller, but this implies smaller confidence level (there is a
trade-off)
• σ is smaller, no control over this.
• n gets larger. Increasing the sample size reduces the margin of error
for any confidence level. *We must quadruple the sample size in
order to cut the margin of error in half.
Example 5
Use the data regarding tip percentages. The 95% confidence interval for the mean
percentage tip for all patrons of the restaurant when their bill contains a message
that the next day’s weather will be good is…
The 90% confidence interval based on the same data replaces the 95% critical
value z = 1.960 with the 90% critical value z = 1.645. This interval is
Lower confidence results in a smaller margin of error. ±0.74 instead of ±0.88. The
margin of error for 99% confidence is larger, ±1.15. Also, if we had a sample of
only 10 patrons, the margin of error for 95% confidence increases from ±0.88 to
±1.24. Cutting the sample size in half does not double the margin of error.
Larger Confidence = Larger Span
Example 6 (exercise 14.8)
Example 7 (exercise 14.19)