significancea; the values 0.01 and 0.05 are common choices fora. Since a test
of levelalphais also of levela^0 >a, it is convenient to give a name to the
smallest level of significance of a test. This quantity is called thesizeof the test.
In general, if we have a test statisticTand use a critical valuet, our test has size
aðtÞ, which depends on the value oft, given by
aðtÞ¼PrðTbtÞThe problem is that di¤erent people faced with the same testing problem
may have di¤erent criteria of size; investigator A may be satisfied to reject the
null hypothesis using a test of size 0.05, while investigator B insists on using
size 0.01. If the two investigators can agree on a common test statisticT, this
di‰culty may be overcome by reporting the outcome of the investigation in
terms of theobserved sizeorpvalue of the test. In other words, thepvalue isa
(t¼observedT). This quantity is a statistic that is defined as the smallest level
of significance at which an investigator usingTwould rejuect on the basis of
the outcome observed. That is, if the investigator’s critical value corresponds to
a test of size less than thepvalue, the null hypothesis is rejected. If the null
hypothesis is true and the investigation repeated,pvalues are distributed uni-
formly on the interval½ 0 ; 1 .
EXERCISES
5.1 For each part, state the nullðH 0 Þand alternativeðHAÞhypotheses.
(a)Has the average community level of suspended particulates for the
month of August exceeded 30mg/cm^3?
(b)Does the mean age of onset of a certain acute disease for school-
children di¤er from 11.5?
(c)A psychologist claims that the average IQ of a sample of 60 children
is significantly above the normal IQ of 100.
(d)Is the average cross-sectional area of the lumen of coronary arteries
for men, ages 40–59, less than 31.5% of the total arterial cross sec-
tion?
(e)Is the mean hemoglobin level of high-altitude workers di¤erent from
16 g/cm^3?
(f)Does the average speed of 50 cars as checked by radar on a particu-
lar highway di¤er from 55 mph?
5.2 The distribution of diastolic blood pressures for the population of female
diabetics between the ages of 30 and 34 has an unknown meanmand a
standard deviation ofs¼9 mmHg. It may be useful to physicians to
know whether the meanmof this population is equal to the mean dia-
stolic blood pressure of the general population of females of this age204 INTRODUCTION TO STATISTICAL TESTS OF SIGNIFICANCE