- The Power Curve for Two-sided Alternative Hypotheses
You should complete this assignment using Excel.
- Using the data on heights
we collected from the class, calculate and plot the
power curve for the mean height of all students (males
and females together). Suppose the null hypothesis
for the population mean is your personal height in
inches. (Recall, everyone self-reported their
height at the
beginning of the semester.) The range of
alternative hypotheses should match the range of the
actual data we collected. Use alpha = 5% as the level
of significance (two-sided). Work under the
assumption that the height data is not distributed
normally and that the variance is unknown.
Steps: (Print out one page with the answers to all the steps
below, and a second page with the requested graph.)
- Calculate the standard error of the mean for our sample data.
- Calculate the lower and upper end points of the
acceptance region (in inches) under the null hypothesis.
(Hint: consider the
NORMINV function in Excel.)
- Let your alternative hypothesis be that the average height
of the class is some other height in the range of the class.
Find the probability under this particular alternative
hypothesis that the sample mean will lie below the
numerical value of the lower end point identified above.
(Hint: consider the
NORMDIST function.)
- Find the probability under the same alternative
hypothesis that the sample mean will lie above the numerical
value of the upper end point identified above.
(Note: the
NORMDIST function gives a left-hand tail only.)
- What is the power of the test for this particular alternative?
(Hint: it should be a function of the answers to (3) and (4) above.)
- Repeat (3) - (5) for all the possible alternative hypotheses
that lie in the range of our original data. Display your results,
clearly and concisely, as a table. (Hint: Use one-inch increments,
from the minimum to the maximum height for the class.
If you set up the previous steps appropriately, all you should
need to do is cut and paste a few formulas to create this table.)
- Plot the points of the power curve you have calculated in
(6) as a line graph.
- Choose five student heights at random (with replacement):
list them and
calculate a sample mean. Does that sample mean lie within
the 95% confidence interval implied by your end points in (2)?
Calculate the two-sided p-value for your sample mean.
- Repeat steps (1) - (7) above for a sample size of five, rather
than the full class as above. Use the sample mean you computed
in (8) above as the value for the population mean under the
null hypothesis.
Notice that the only statistical difference between this
question and the one above concerns the shape of the distribution.
(Hint: with some judicious design of the spreadsheet used to
answer part (A),
you can simply copy and paste the above results, make a few small
changes, and get the desired answers for (B).) Then:
- Choose at random (with replacement) a second group of
five student's heights:
list them and calculate this sample's mean. Does this
sample mean height lie within the 95% confidence interval
you have computed for this question? Calculate the
two-sided p-value for this sample mean height under your
null hypothesis.
(Print out a third page with the answers to all the steps of part (B),
and a fourth with the graph of the power curve for (B).)
- Compare the power curves (power functions) computed in questions
(A) and (B) above.
- What is the maximum value of the power function for (A)?
What is the maximum value of the power function for (B)?
Explain why each obtains the maximum value you computed.
- What is the minimum value of the power function for (A)?
What is the minimum value of the power function for (B)?
Explain why each obtains the minimum value you computed.
(Print out a fifth page with the answers to these questions.)
- Data files for these questions:
The raw data for this problem is available as
an Excel file;
click on the link to download.
(Ask a lab consultant if you need help.)
- Submitting this assignment for credit:
-
In order to receive credit for this assignment,
you must also e-mail to me your Excel spreadsheet
as an attachment.
Please name the Excel file as your e-mail name
-- e.g.
mshanson.xls -- and include
your name in a cell of the worksheet as well.
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