Names:
:
Confidence, Significance and P-Values
Work in groups of 2 or 3 (turn in only one paper per group). We will learn
how to do significance tests with the TI-83 calculators in this activity.
We start with an example done by hand.
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Acute myeloblastic leukemia is among the most deadly of cancers. Experience
indicates that the time in months that a patient survives after initial
diagnosis of the disease is normally distributed with a mean of 13.0 months
and a standard deviation of 3 months. A new treatment is being investigated
that should prolong the average survival time without affecting the variability.
The survival times for 16 patients receiving the new treatment are listed
below.
| 17.8 |
12.5 |
16.7 |
17.3 |
12.5 |
18.3 |
10.5 |
13.3 |
| 21.4 |
16.2 |
10.3 |
11.8 |
8.8 |
13.2 |
12.5 |
15.6 |
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Set up a significance test for the information above by defining the null
hypothesis H0 and the alternative hypothesis Ha.
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Calculate the mean,
,
of the sixteen survival times.
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Using the fact that the population standard deviation is equal to 3, what
is the standard deviation of the sampling distribution for
(assuming the null hypothesis is true)?
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Calculate the p-value of your significance test for the observed value
of .
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Based on your p-value, do you think that the observed value of
would lie in a 95% confidence interval (for the statistic
= mean of 16 survival times) centered at the population mean µ =
13? Explain.
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Find a 95% confidence interval centered at 13 for the statistic .
Using the Z-Interval command, set s =
3 (the population standard deviation)
= 13, n = 16 and C-Level = .95. Is the observed value of
in this interval?
Now we'll repeat the significance test above with some new features
of the TI-83. Most likely you already calculated the values of
using the 1-VarStats feature of your calculator (so the data is
already in L1). Be sure that the 16 times are stored in the
list L1 and delete or turn off all Y-variable plots and STAT
plots. We can calculate the p-value of a significance or hypothesis test
quite quickly with the TI-83. This can be done either with data (such as
the list of survival times), or with statistics (the values of x-bar and
n = the sample size). For both methods you'll need to know the population
mean, µ, and the population standard deviation, s.
Press [STAT] and then select the [TESTS] submenu. Hypothesis tests (of
this type anyway) are called Z-Tests on the TI-83 and they correspond to
the first choice on the [TESTS] submenu. Select 1:Z-TEST. The hypothesis
test editor screen should appear. You move around with the arrow keys and
make selections by pressing [ENTER]. Select DATA for the input type.
Enter the population mean (13) for µ0 and the population
standard deviation (3) for s. Give L1
as the list and set the frequency to 1. The second to last line is where
you select the form of the test. You can chose from
a two tailed test 
a left tailed test 
or a right tailed test 
Select the right tail test icon and then select DRAW on the bottom line
to see a plot and the p-value. The calculator plots the standardized version
of the problem and reports the standardized z-value as well.
What is the z-value reported?
Did you get the same p-value as above?
Now, go back to the Z-Test screen and select STATS this time
for the input type. The calculator should display the screen below.
The values from your last test should remain in the Z-Test editor. Move
to the bottom line and select Calculate this time. You should get
the same p-value and the same Z-value as before, just no graph this time.
The calculate option is nice if you only need a number for the p-value
or z-value. Here is one more example to try out using the new commands.
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Under normal conditions, annual growths of a certain type of tree are known
to be normally distributed with a mean of 11 inches of growth per year
and a standard deviation of 2.3 inches. An experimental insecticide is
used on a sample of 48 of these trees, and the researcher is interested
in seeing if the insecticide has any effect on the annual growth of the
trees.
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Does this setting call for a right-tailed test, left-tailed test or two-tailed
test? Why?
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Define the appropriate null and alternative hypothesis for this significance
test.
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The researcher finds that with the insecticide, the 48 trees grew an average
of 10.3 inches. Assume the insecticide has no effect on the standard deviation
of these growths. Draw a graph that illustrates the p-value (as an area
under a normal density curve) and report the p-value of this test.
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Explain in everyday language, how confident you are that the insectide
did or did not affect the tree's growth.