Download December 04, 2012 10.2 - Confidence Intervals for the True Mean, µ

December 04, 2012
10.2 - Confidence Intervals for the True Mean, µ
Just like we have a confidence intervals to determine
where the true proportion of a population lies for a piece
of given data, we also have a confidence interval to
determine where the true mean of a population lies.
December 04, 2012
The figure illustrates these facts about the t-distributions:
1) The density curves of the t distributions are similar in shape to the standard
Normal curve. They are symmetric about zero, single-peaked, and bell-shaped.
2) The spread of the t distributions is a bit greater than that of the standard Normal
distribution. The t distributions in Figure 10.7 have more area (probability) in the
tails and less in the center than does the standard Normal.
3) As the degrees of freedom increase, the t-density curves approach the standard
Normal distribution. This happens because s estimates σ more accurately as the
sample size increases. So using s in place of σ causes little extra variation when the
sample is large. The center is µ.
December 04, 2012
Example 10.14 - Measuring Stream Health
The level of dissolved oxygen in a river is an important indicator of the water’s
ability to support aquatic life. A researcher collects water samples at 15 randomly
chosen locations along a stream and measures the dissolved oxygen. Here are the
results in milligrams per liter:
4.53 5.04 3.29 5.23 4.13
5.50 4.83 4.40 5.42 6.38
4.01 4.66 2.87 5.73 5.55
We will construct and interpret a 95% confidence interval for the mean dissolved
oxygen level in this stream.
What is µ in this setting? It’s the mean dissolved oxygen level in the stream. In
order to use a t-interval to estimate µ, we need to know that the population
distribution is Normal or that there are no outliers or strong skewness in the data.
Exercise: input the above data into your
calculator and generate a box plot to determine
if there are any outliers or strong skewness.
December 04, 2012
To compute the confidence interval on your calculator....
Step 1: Create a list for your data
You must assign a
variable to your list.
You will need this
further down in the
process!!
Step 2: Find confidence interval for your data. In the MENU, perform
the following sequence of steps: Open up a calculator page first.
Conclusion: We are 95% confident that the true mean
DO level in the stream lies within 4.25 and 5.29 mg/1.
December 04, 2012
Finding t*
In order to determine what t* is, here is the minimum
amount of information you need:
1) Area under the normal curve (or p-value). You
get this information from knowing the confidence
level.
2) The Degrees of Freedom (n - 1)
3) Recall how we found z* given p-value, mean, and
standard deviation for normal distributions earlier in
the semester? We can find t* in the same way for
sampling distributions because the curve of sampling
distributions are similar to the shape of normal
density curve.
invt (area under normal curve, df )
Example: What critical value t* should be used for a 99%
confidence interval for the population mean, µ, based on
n = 15 observations?
invt (.995, 14)
Answer:
December 04, 2012
Example: What standard deviation would I need in order to achieve a
margin of error of .05 in a sampling distribution with a level
of confidence of 92% and a sample size of 20?