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Grading
Your grade in the course will be determined from three hour tests, counting 100
points each, and a final exam, counting 200 points.
If you have a Java capable browser you can utilize this section.
- Select a random sample, without replacement,
of your desired sample size from the numbers 1 to a population size you set. After
entering the population size and sample size click on the "sample without replacement" button. The
program will then select a random sample to your specifications and you can see all the numbers by
hitting reload on your browser and scrolling, or you can save them to a file (on the Mac) by putting the
cursor in the large white area, selecting all, and copying.
- You can "look up" a standard normal probability; just enter the desired z and press the calculate
button.
- Plot normal densities . You can see plotted on
the same axes a standard normal density in black and a normal density of your choice in blue. Select a
standard deviation by clicking along the horizontal line segement at the top; select a mean by clicking
along the horizontal axis.
- The tests given on the hypothesis testing sheet are known to perform well and to prove it
mathematically takes considerable effort. A good understanding of what makes these tests work can be
had with much less effort as in the following example. Suppose a random sample of n = 5 observations is
taken from a normal population and we are interested in testing the null hypothesis that the standard
deviation is some particular value, which we shall call here the null standard deviation. The
appropriate test statistic is (n-1)sample variance/null variance and if
the null hypothesis is true, it follows a chi-square distribution with n-1 = 4 degrees of freedom.
However, if some alternative value other than our null value is the correct population standard
deviation,then the test statistic does not have a chi-square density. You can select the value of the null
and alternative standard deviations and see on the same axes
plots of the chi-square density (plotted in black and correct under the null) and the actual
density of the test statistic plotted in red ( the correct one under the true alternative value you
selected). You will see that as you move the true population standard deviation away from the
hypothesized value the test statistic's distribution is concentrated increasingly in one or the other of
the two tails of the null distribution. In the next section this behavior is investigated more thoroughly
in a different means example through the power function of the test. (If you are using a Mac the applet
above may not work correctly, however you can use the
Mac version. Just click along the axis to select the ratio of variances.)
- In testing statistical hypothesis the probability (called the power of the test ) of rejecting a null
hypothesis depends on the true distribution and on the rejection region (decision rule). In this example
we investigate the power function of a test of whether the mean of a normal, with known standard
deviation 16, is 0 against the alternative that the mean is not 0, based on a single observation. It can be
shown that the optimal rejection region whose size is 0.0455 is to reject the null hypothesis if | x |
exceeds 32. You can explore the power function of this test
as a function of the mean of the true normal distribution; just click the mouse along the axis to select a
mean. You will see below the graph the mean you selected and the power (probability of rejecting the
null hyppothesis) for your selection. The red curve is a plot of the corresponding normal density and the
area of the red portion is the power.
- You can explore p-values for hypothesis tests of a
mean based on a random sample of 4 observations from a normal distribution. We know that the tests of
the mean of a single normal population which are found on the testing sheet can be recast in terms of p-
values to read: reject the null hypothesis at level alpha if the p-value of the data is smaller than
alpha. In the applet you can select the alternative hypothesis and find the p-value of the data by the
clicking the mouse along the x-axis to set an observed value of the test statistic (a t statistic in this case
of course). The black curve is a graph of the t-density with 3 degrees of freedom, which is the correct
probability distribution of the test statistic under the null hypothesis, and the area of the shaded
portion is the p-value of the particular value of the test statistic which you selected.
If you have release 4 of Maple V on your machine you can utilize this section.
Simply make sure the settings on your web browser under options:helper applications are MIME
Type application/Maple and suffixes are mws,ms.
- Simulate Weibull random variables and see
plots of the histograms of the results.
- In class we proved mathematically that the square of a standard normal random variable has a
chi-square distribution with one degree of freedom. Empirical evidence of this can be obtained here. Simulate n standard normal random variables, forming a
histogram of their squares, and plotting on the same axis the chi-square 1 density. You can try different
values of n. For n a thousand or so you'll observe a pretty close correspondence.
- Simulate an instance of the central limit theorem.
- Plot histograms of data. First put the file
containing the data of interest, an ordinary text file named "somedata," in your Maple folder (or
directory). The data should be delimited by spaces or EOL's. There are some sample data sets below
which you can try, just select the one of interest. You can easily save the file as required from your
browser. Similar instructions apply to see a boxplot of
your data.
Data sets in the correct format are listed below. Other data sets
can be obtained at remote sites but most require pre-processing.
- Results of a simulation of 50 normal random variables of
mean 5 and variance one-
- Times to failure of Kevlar 49/epoxy, a material used in
the space shuttle-
- Results of a simulation of 100 Poisson random variables
with mean 6-
- The data set of concentration of suspended solids
in river water from exercise 1.10 of Devore-
If you have a spreadsheet on your machine which will read SYLK files you can utilize this
section.
The settings on your web browser under options : helper applications are MIME Type
text/spreadsheet and the suffix is slk. Just set your browser to open your spreadsheet with these
settings. When you do this you will have a working spreadsheet, formulas and all, which you can
easily adapt to analyze other data similarly by simply pasting in the data part from a text file.
- A spreadsheet can be obtained which computes
the average and standard deviation of the data set from exercise 1.10 of Devore mentioned above.
- A spreadsheet with data from a study of the
retention, in mice, of two forms of orally administered dietary iron supplements. Assuming normality,
the confidence interval is given for the difference in the mean retention of the two types.
- A 1968 paper by Linus Pauling spurred interest in the role of vitamin C in mental illness. An
experiment was performed to study this relationship. In the study 15 schizophrenics and 15 patients
with diagnoses of neuroses of a different origin were selected and had their intake of ascorbic acid
carefully controlled for four weeks before being given a dose of 1 g of ascorbic acid. Urinary excretions of
vitamin C of patients were kept over a span of six hours after administration and the amounts of
vitaminC are found in the data sheet. Does the data present significant evidence that the mean
excretions of vitamin C for the two types of patients differ ? This
spreadsheet contains the data and some computations. The
vitamin C data can be obtained as a text file even if you don't have a spreadsheet. The first
column contains the numbers for the schizophrenics and the second the numbers for the controls.
