Download Estimating with Confidence

It is commonly believed that anyone who tabulates numbers is a statistician.
This is like believing that anyone who owns a scalpel is a surgeon. – Hooke
Chapter 11 - Sec. 11.1 - Inference for a Population Mean
The last chapter provided practice finding confidence intervals and carrying out tests of
significance in a somewhat unrealistic setting. We needed the population σ. In reality
σ and μ are rarely known.
The conditions for inference about a mean are as before: SRS and a normal distribution.
For these smaller (n<30) samples ALMOST normal is actually good enough as long as
the data are mostly symmetric without multiple peaks or outliers. Smaller samples are
better handled with a t distribution since normality cannot be validated using the Central
Limit Theorem.
The new t distribution:
When we don’t know σ, we have to have a new standard deviation because

won’t
n
work. So we replace σ with s, the SAMPLE standard deviation and call it the standard
s
error. Standard Error (S.E.) =
. Instead of using the normal distribution, we are now
n
working with the t distribution. The graph of a t distribution is similar to a normal
density curve: symmetric about 0, single peaked, and bell-shaped. The spread is a bit
wider giving more probability in the tails and less in the center. (More variation is
present when using s in place of σ. As the df* increase the density curve gets closer and
closer to N(0,1)).
*A new term must be considered when working with t distributions and that is “degrees
of freedom” df– simply put DEGREES OF FREEDOM = n – 1. (NOTE: there is a
different t value for different sample sizes since small samples have high variability.)
We will have to adjust to these changes by using Table C (t-distributions) on the inside
back cover of the textbook to determine critical t* values. (See p 618-19 for picture and
explanation on how to use Table C)
NEW PROCEDURES FOR C.I. AND SIGNIFICANCE TESTS:
-We will now be doing a one sample mean t C.I. or a one sample mean t test whenever
we do not know σ.
-Assumptions: We still need an SRS. If n > 30, CLT applies. If n < 30, plot the sample
data. If there are no major outliers and it is fairly symmetric, then we proceed.
s
x 
Confidence Interval: x  t * 
Test: t 
df = n - 1
s
n
n
Use Table C to find the t* for the C.I. and the p-value for the test. You must multiply the
p-value by two for a two tailed test.
TI-83/84: STAT – TESTS – 2: T-Test or 8: T Interval
See Example 11.2 on p 623
s
is often referred to as the Standard Error (S.E.) or standard error of the
n
mean (S.E.M.)
Example: A medical study finds that x = 114.9 and s = 9.3 for the seated systolic blood
pressure of 27 members of one treatment group. What is the standard error of the mean?
s
9.3
SE =
=
= 1.789
n
27
NOTE:
Robustness of t procedures:
A C.I. or test is called robust if the confidence level or p-value does not change very
much when the assumptions of the procedure are not met.
1) t procedures are quite robust against nonnormality of the population when there are no
outliers, especially, when the distribution is roughly symmetric. The t procedures are
strongly influenced by outliers.
2) Always make a plot to check for skewness and outliers before you use the t
procedures. For most procedures, you can safely use the t test or t C.I. when n is at least
15 unless an outlier or strong skewness is present.
Matched pairs t procedures:
To compare the responses to the two treatments in a matched pairs design, apply the one
sample t procedures to the observed differences. The parameter, μ, in a matched pairs t
procedure is the mean difference in the responses to the two treatments within matched
pairs of subjects in the entire population.
Put one set of data in L1 and the other set of data in L2. We then need to find their
DIFFERENCES so go to the very top of L3, highlight L3, and type in L1 – L2. Now
perform a 1-Var STATS on L3. This will give you your sample mean difference, xd , and
your sample standard deviation, sd, which are needed for the formulas.
New Step 1:
μd = the true mean difference in…
New Step 2:
H0: μd = 0
(there is no difference)
Ha: μd ≠ 0 or μd < 0 or μd > 0
Let’s look at the 1997 AP exam question for an example 