They form the core of the expanded table.
There were 11 girls with low heart rate, 17 boys with low heart rate, and so on. The four entries in boldface type are counts of observations from the sample of n = 40. By adjoining row totals, column totals, and a grand total we obtain the table shown as Table 11.1 "Baby Gender and Heart Rate". The 40 records give rise to a 2 × 2 contingency table. A heart rate below 145 beats per minute will be considered low and 145 and above considered high. We divide the second factor, heart rate, into two levels, low and high, by choosing some heart rate, say 145 beats per minute, as the cutoff between them. The factor gender has two natural categories or levels: boy and girl. H a : Baby gender and baby heart rate are n o t independent Since the burden of proof is that heart rate and gender are related, not that they are unrelated, the problem of testing the theory on baby gender and heart rate can be formulated as a test of the following hypotheses: H 0 : Baby gender and baby heart rate are independent vs. In this context these two random measures are often called factors A variable with several qualitative levels. We examine the heart rate records of 40 babies taken during their mothers’ last prenatal checkups before delivery, and to each of these 40 randomly selected records we compute the values of two random measures: 1) gender and 2) heart rate. There is a theory that the gender of a baby in the womb is related to the baby’s heart rate: baby girls tend to have higher heart rates. We build the discussion around the following example. Thus the hypotheses will be expressed in words, not mathematical symbols. In this subsection we will investigate hypotheses that have to do with whether or not two random variables take their values independently, or whether the value of one has a relation to the value of the other. Hypotheses tests encountered earlier in the book had to do with how the numerical values of two population parameters compared. arises in tests of hypotheses concerning whether or not two population variances are equal and concerning whether or not three or more population means are equal. The F-distribution A particular probability distribution specified by two degrees of freedom, d f 1 and d f 2. arises in tests of hypotheses concerning the independence of two random variables and concerning whether a discrete random variable follows a specified distribution. The chi-square distribution A particular probability distribution specified by a number of degrees of freedom, d f. Whereas the standardized test statistics that appeared in earlier chapters followed either a normal or Student t-distribution, in this chapter the tests will involve two other very common and useful distributions, the chi-square and the F-distributions.
The idea of testing hypotheses can be extended to many other situations that involve different parameters and use different test statistics.
In previous chapters you saw how to test hypotheses concerning population means and population proportions.