Every test of significance begins with a * null hypothesis H_{0}*.

The * alternative hypothesis*,

The final conclusion once the test has been carried out is always given in terms of the null
hypothesis. We either "reject *H _{0}* in favor of

If we conclude "do not reject *H _{0}*", this does not necessarily mean that the null
hypothesis is true, it only suggests that there is not sufficient evidence against

(*Definitions taken from Valerie J. Easton and John H. McColl's
Statistics Glossary v1.1*)

Hypotheses are always stated in terms of population parameter, such as the mean
.
An alternative hypothesis may be * one-sided* or

Hypotheses for a one-sided test for a population mean take the following form:
*H _{0}*: = k

Hypotheses for a two-sided test for a population mean take the following form:
*H _{0}*: = k

A ** confidence interval** gives an estimated range of values
which is likely to include an unknown population parameter, the estimated
range being calculated from a given set of sample data.
(

The null hypothesis *H _{0}* claims that there is no difference between the mean
score for female students and the mean for the entire population, so that
= 70. The alternative hypothesis claims that the mean for
female students is higher than the entire student population mean, so that
> 70.

The test statistic follows the standard normal distribution (with mean = 0 and standard deviation
= 1). The test statistic *z* is used to compute the *
P-value* for the standard normal distribution, the probability that a value at least as
extreme as the test statistic would be observed under the null hypothesis. Given the null
hypothesis that the population mean is equal to a given
value

The probability is doubled for the two-sided test, since the two-sided alternative hypothesis
considers the possibility of observing extreme values on *either* tail of the normal
distribution.

z = (73 - 70)/(10/sqrt(64)) = 3/1.25 = 2.4. Since this is a one-sided test, the

In a one-sided test, corresponds to the critical value
*z ^{*}* such that

Another interpretation of the significance level ,
based in * decision theory*, is that corresponds
to the value for which one chooses to reject or accept the null hypothesis

Since the pharmaceutical company is interested in *any* difference from the mean recovery
time for all individuals, the alternative hypothesis *H _{a}* is two-sided:
30. The test statistic is calculated
to be

Decision theory is also concerned with a second error possible in significance testing, known as

=

=

**For claims about a population mean from a population with a normal
distribution or for any sample with large sample
size n (for which the sample mean will follow a normal distribution by the
Central Limit Theorem) with unknown standard deviation,
the appropriate significance test is known as the t-test, where the test
statistic is defined as t = .**

The test statistic follows the *t* distribution with *n-1* degrees of freedom.
The test statistic *z* is used to compute the *
P-value* for the

The dataset "Normal Body Temperature, Gender, and Heart Rate" contains 130 observations of body temperature, along with the gender of each individual and his or her heart rate. Using the MINITAB "DESCRIBE" command provides the following information:

Descriptive Statistics Variable N Mean Median Tr Mean StDev SE Mean TEMP 130 98.249 98.300 98.253 0.733 0.064 Variable Min Max Q1 Q3 TEMP 96.300 100.800 97.800 98.700Since the normal body temperature is generally assumed to be 98.6 degrees Fahrenheit, one can use the data to test the following one-sided hypothesis:

*H _{0}*: = 98.6 vs

The *t* test statistic is equal to (98.249 - 98.6)/0.064 = -0.351/0.064 = -5.48.
*P(t < -5.48) = P(t> 5.48)*. The

To perform this *t-test* in MINITAB, the "TTEST" command with the "ALTERNATIVE" subcommand
may be applied as follows:

MTB > ttest mu = 98.6 c1; SUBC > alt= -1. T-Test of the Mean Test of mu = 98.6000 vs mu < 98.6000 Variable N Mean StDev SE Mean T P TEMP 130 98.2492 0.7332 0.0643 -5.45 0.0000These results represents the exact calculations for the

*Data source: Data presented in Mackowiak, P.A., Wasserman, S.S., and Levine, M.M. (1992),
"A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and
Other Legacies of Carl Reinhold August Wunderlich," Journal of the American Medical
Association, 268, 1578-1580. Dataset available through the
JSE Dataset Archive.*

Analysis of data from a matched pairs experiment compares the two measurements
by subtracting one from the other and basing test hypotheses upon the
differences. Usually, the null hypothesis *H _{0}* assumes
that that the mean of these differences is equal to 0, while the alternative
hypothesis

In MINITAB, subtracting the air-filled measurement from the helium-filled measurement for each trial and applying the "DESCRIBE" command to the resulting differences gives the following results:

Descriptive Statistics Variable N Mean Median Tr Mean StDev SE Mean Hel. - Air 39 0.46 1.00 0.40 6.87 1.10 Variable Min Max Q1 Q3 Hel. - Air -14.00 17.00 -2.00 4.00Using MINITAB to perform a

T-Test of the Mean Test of mu = 0.00 vs mu > 0.00 Variable N Mean StDev SE Mean T P Hel. - A 39 0.46 6.87 1.10 0.42 0.34The P-Value of 0.34 indicates that this result is not significant at any acceptable level. A 95% confidence interval for the

*Data source: Lafferty, M.B. (1993), "OSU scientists get a kick out of sports controversy,"
The Columbus Dispatch (November 21, 1993), B7. Dataset available through the
Statlib Data and Story Library (DASL).*

**To perform a sign test on matched pairs data, take the difference between the two measurements
in each pair and count the number of non-zero differences n. Of these, count the number
of positive differences X. Determine the probability of observing X positive
differences for a B(n,1/2) distribution, and use this probability as a P-value
for the null hypothesis.**