10-2
Estimating a Population Mean
(σ Unknown)
Warm-up Question
•High School students who take the SAT Mathematics
exam a second time generally score higher than on
their first try. The change in the score has a Normal
distribution with standard deviation σ=50. A random
sample of 250 students gain on average x-bar=22
points on their second try.
•Construct a 95% Confidence interval for μ
•How large a sample of high school students would
be needed to estimate the mean change in SAT score
μ to within ± 2 points with 95% confidence?
Confidence Intervals Involving Z
Using the Calculator
•Because we don’t know σ, we estimate it by
using the sample standard deviation, s.
•We then estimate the standard deviation of xbar using s / sqrt(n)
•This is called the standard error of the
sample mean x-bar
“Standard error”: You are estimating the
standard deviation…but there will likely be
some error involved because we are
estimating it from sample data.
In other words… the standard error is (most
likely) an inaccurate estimate of a (population)
standard deviation.
The t distributions
When we substitute the standard error of xbar (
s/sqrt(n) )for its standard deviation ( σ/sqrt(n)),
the distribution of the resulting statistic, t, is not
Normal.
We call it the t distribution.
The t-statistic was introduced in 1908 by William
Sealy Gosset, a chemist working for the Guinness
brewery in Dublin, Ireland ("Student" was his pen
name). Gosset devised the t-test as a way to
cheaply monitor the quality of stout.
The t distributions
There is a different t-distribution for each sample
size n.
We specify a t distribution by giving its degrees of
freedom, which is equal to n-1
We will write the t distribution with k degrees of
freedom as t(k) for short.
We also will refer to the standard Normal
distribution as the z-distribution.
Comparing t and z distributions
Compare the shape,
center, and spread of
the t-distribution with
the z-distribution.
As the degrees of freedom k increase, the t(k)
density curve approaches the N(0,1) curve ever
more closely. As the sample size increases, s
estimates σ ever more closely.
Finding t with Table C
Suppose you want to construct a 95% confidence
interval for the mean μ of a population based on a
SRS of size n=12. What critical value t should
you use?
Finding t with Table C
Suppose you want to construct a 90% confidence
interval for the mean μ of a population based on a
SRS of size n=15. What critical value t should
you use?
Suppose you want to construct a 80% confidence
interval for the mean μ of a population based on a
SRS of size n=5. What critical value t should you
use?
Our formula is the
same as it was for zintervals EXCEPT we
replace sigma with s!!!
One sample t interval for mu
Recall the inference tool-box
One sample t interval for mu
Environmentalists, government officials, and
vehicle manufacturers are all interested in
studying the auto exhaust emissions produced by
motor vehicles. The table gives the nitrogen oxide
(NOX) levels for a random sample of light-duty
engines of the same type.
One sample t interval for mu
Construct a 95% confidence interval for the mean
amount of NOX emitted by light-duty engines of
this type.
One sample t interval for mu
Step 1: Parameter
Step 2: Conditions
Step 3:Calculations
One sample t interval for mu
Step 1: Parameter
Step 2: Conditions
Step 3:Calculations
Step 4:Interpretation
Remember the three C's! Conclusion, Connection, Context
One sample t interval for mu
Note: When the actual df does not appear in
Table C, use the greatest df available that is less
than your desired df.
Ronald McDonald’s sister Diana Rhea is the
purchasing manager for domestic hamburger
outlets. The company has decided to provide a
free package of Tums to any complaining
customer. In order to estimate monthly demand,
she took a sample of 5 outlets and found the
number of Tums distributed to customers in a
month was
250, 280, 220, 280, 320
(a)Find the sample mean and sample standard
deviation
(b)Construct a 90% confidence interval on the
average monthly demand per outlet.
Homework!