127.3 
127.3
Rweb in Stat 3011 Home Page Stat 3011 Home Page About the Rweb in Stat 3011 Web Pages
R has no simple procedure for doing "large sample" procedures for means. The reason is that when using a computer, you might as well use the "small sample" procedure. Just ignore the n > 30 rule and use the t distribution for all procedures. It will do almost the same thing for large n and be more conservative to boot. Hence, to do things the easy way, go to the small sample section.
To do the "large sample" confidence interval as described in the textbook, you must implement the formulas on the computer. For example, consider Example 8.6 in Devore and Peck. The problem data are
| sample mean | 5.14 |
|---|---|
| sample s. d. | 1.30 |
| sample size | 52 |
1.96 * 1.30 / sqrt(52)
for which Rweb gives 0.353344. Hence the interval is
5.14 ± 0.353.
If we like, we can use qnorm to look up the z
critical value. The probability we want to look up is .025 because
there is 2.5% in each tail. Thus
- qnorm(0.025) * 1.30 / sqrt(52)
gives a slightly more precise answer.
If you want the endpoints of the interval you can do
5.14 + qnorm(0.025) * 1.30 / sqrt(52)
5.14 - qnorm(0.025) * 1.30 / sqrt(52)
or the same thing in one statement with
5.14 - c(-1,1) * qnorm(0.025) * 1.30 / sqrt(52)
Again, we just use Rweb as a calculator. This time we use Example 9.12 in Devore and Peck. The problem data are
| sample mean | 130 |
|---|---|
| sample s. d. | 8 |
| sample size | 101 |
127.3 127.3
(130 - 127.3) / (8 / sqrt(101))
which Rweb says is 3.391833. The one-tailed P-value is the
tail area to the right of this
z <- (130 - 127.3) / (8 / sqrt(101))
1 - pnorm(z)
Of course, this particular example asks for a two-tailed test, and
a two-tailed P-value is twice the one-tailed
z <- (130 - 127.3) / (8 / sqrt(101))
2 * (1 - pnorm(z))
If the test statistic had turned out negative, the one-tailed P-value for a lower-tailed test would be
pnorm(z)
and the two-tailed P-value would be
2 * pnorm(z)
Single expressions that to either job, upper-tailed or lower-tailed are
pnorm(- abs(z))
for the one-tailed P-value, and
2 * pnorm(- abs(z))
for the two-tailed P-value. Question: why do these work?
You should figure this out.
For this section, we will use the data set for Example 8.10 in Devore and Peck
http://superior.stat.umn.edu/~charlie/3011/te0810.dat
Rweb has a function t.test that does t tests and
confidence intervals. Type the URL above in the data URL window and
submit
t.test(x)
and Rweb returns, among other things, the 95% confidence interval
(0.8876158, 0.9633842).
This example, actually asks for 99% confidence rather than 95% confidence. To get that you use an optional argument (all the optional arguments are described on the on-line help).
t.test(x, conf.level=0.99)
gives the answer in the book.
For this section, we will use the data set for Problem 9.57 in Devore and Peck
http://superior.stat.umn.edu/~charlie/3011/ex0957.dat
As the name indicates, the
Rweb function t.test does t tests as well as
confidence intervals
Type the URL above in the data URL window and
submit
t.test(x, mu=25)
using the optional argument mu to specify the null hypothesis.
By default Rweb does a two-tailed test, that is, the hypotheses are
25 25
This is not the answer in the back of the book for part (b). The book asks for an upper-tailed test, with hypotheses
25 
25
t.test(x, mu=25, alternative="greater")
this gives P = 0.04263 for the P-value, which agrees
with the answer in the back of the book. You can find the optional argument
for any eventuality by looking at the
on-line help).
No fuss. You don't have to calculate anything. Rweb does it all.
Unfortunately, when doing problems out of the book, the book usually
gives only the sample mean and standard deviation, which the
t.test function can't use. It wants the actual data.
Thus Rweb is fairly useless for doing textbook toy problems, but
very useful for doing real-world problems.
If you want to do a textbook problem with no data, you will have to do
it like the examples in the large sample section
replacing the pnorm and qnorm functions with
the
pt and qt functions, which calculate probabilities
and quantiles for the t distribution, see the
on-line help for details.
As for the t-test, there is an Rweb function prop.test
that does large sample tests and confidence intervals for proportions.
For a confidence interval example we will use Example 8.8 in Devore and Peck. The problem data are
| x | 466 |
|---|---|
| n | 1014 |
prop.test(466, 1014)
Rweb's confidence interval (0.4286132, 0.4908291) agrees with the book.
As with all Rweb functions, see the
on-line help for details of how to use it.
For a hypothesis test example we will use Example 9.15 in Devore and Peck. The problem data are
| x | 51 |
|---|---|
| n | 300 |
0.25 
0.25
prop.test(51, 300, p=0.25, alternative="less")
and Rweb gives the one-tailed P-value P = 0.0008642,
which does not agree with the answer in the book P = 0.0007.
What happened?
Actually, the book is wrong. Rweb has gotten a better answer using (as it says in the printout) a "continuity correction". To understand how Rweb actually gets this answer and why it is better than the book's answer, you would have to look in another book, but we won't go into that. You could get the answer in the book from
prop.test(51, 300, p=0.25, alternative="less", correct=FALSE)
(the "correct=FALSE" says it all).