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Stat211/Labpack/ Confidence Intervals for mu
Confidence Interval for m:
Goals of the lab:
•
To have students become familiar with the calculation and interpretation of a confidence interval for
the population mean.
Background:
A (1-a)*100% confidence interval for the population mean is discussed in this lab. The first formula,
given below, is the formula for creating this confidence interval when the size of the sample is relatively
large, say, n ≥ 20.
There are 3 factors that influence the width (and hence the size of a Confidence interval):
1. The confidence level (1-a.)*100% : The width of the confidence interval increases as we increase the
confidence level
2. The sample size n: As the sample size increases the width of the confidence interval decreases.
3. The variability in the sample, sx : The higher the variability in the data the wider the confidence interval
will be.
Relevant Formula:
X ± z*
sx
n
where z is the
(1 -
a
) th quantile of the standard normal
2
and n ≥ 20
Example:
Suppose we are interested in estimating the mean width of all oak trees in Squirrel Creek National Forest.
We sample 85 oak trees from that forest and find that the sample mean width is 13.67 cm, and the sample
standard deviation is 4.82cm. Use a 95% confidence interval to estimate the mean width of all oak trees in
that forest.
X ± z*
sx
n
Is it appropriate to use this formula? Yes, since n = 85 ≥ 20.
Ê 4.82 ˆ
Ë 85 ¯
=13.67 ± 1.960(0.523)
=13.67 ± 1.025
=( 12.645, 14.695)
=13.67 ± 1.96*
(z=1.96 is the (1 -
0.05
) =0.975 quantile of the standard normal.)
2
Thus we are 95% confident that the mean width of all oak trees in the Squirrel Creek National Forest is
between 12.645 and 14.695 centimeters. This confidence comes from the idea of repeated sampling. If we
repeated this process of sampling 85 oak trees and making 95% confidence intervals from each sample then
(approximately) 95% of the confidence intervals would contain m, the population mean.
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Stat211/Labpack/ Confidence Intervals for mu
More Background:
In the situation where we have a small sample size (n<20), s is known, and the distribution of X is
(assumed to be) Normal we can use the formula on the previous page(s) to find the endpoints of our
confidence interval.
However, in the situation where we have a small sample size (n<20), s is unknown, and the
distribution of X is (assumed to be) Normal we must use a slightly different formula to find the endpoints
of our confidence interval. Rather than using a z-score from a standard normal distribution as a factor in the
error term, we must use a t-value from the appropriate t-distribution, i.e., a t-distribution with df=n-1. .
In this lab, students will get exposure to the Normal probability plot. This is a very difficult
concept for students to grasp. The basic idea (in Stat 211 terms) is that we are checking the data to
determine whether or not these observations came from a Normal distribution. We can do that by looking
at a special plot which if it is a straight line indicates that the data seems to come from a Normal
distribution. The problem is that the data can deviate a good bit from a straight line and still feasibly come
from a Normal distribution. This is due to sampling variability. The key idea to get across to students who
ask about this is that if the line can be called almost straight, then we assume the original distribution is a
Normal distribution.
Relevant formula:
X ±t*
sx
a
where t is the (1 - ) th quantile of a t-distribution with n-1 degrees of freedom
n
2
Example:
A random sample of 15 newborn babies at WVU hospitals is obtained and the birth weight of each baby is
measured. The sample mean is computed to be 6.87 lbs. with a sample standard deviation of 1.76 lbs. Use a
95% confidence interval to estimate the mean birth weight of all babies born at WVU hospital.
X ±t*
sx
n
1.76
(t=2.14 is the 0.975 quantile from a t-distribution with 14 d.f.)
15
= 6.87 ± 2.14 * (0.45443)
= 6.87 ± 0.9725
= 6.87 ± 2.14 *
=( 5.8975, 7.8425)
Thus we are 95% confident that the mean birth weight of all babies born at WVU Hospital is between lbs.
5.8975 and 7.8425 lbs.
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