The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

78 CHAPTER 6 Hypothesis Testing

Table 6.1

Daily Average Temperature in Washington, DC , and New

York City

Date Washington, DC ( ° F) New York City ( ° F)

1. January 15 31 28


2. February 15 35 33


3. March 15 40 37


4. April 15 52 45


5. May 15 70 68


6. June 15 76 74


7. July 15 93 89


8. August 15 90 85


9. September 15 74 69


10. October 15 55 51


11. November 15 32 27


12. December 15 26 24

the course of the year, temperatures in DC range from 93 ° F to 26 ° F, a
difference of 67 ° F, and in New York, from 89 ° F to 24 ° F, a difference
of 65 ° F. But the difference in mean temperature between New York
and Washington ranges only from 2 to 5 ° F. However, New York is
always lower than DC each date of matching.
This is a clear case where this small difference would not be detect-
able with a two - sample (independent samples) t - test. But it would
be easily detected by a paired t - test or a nonparametric approach
(sign test).

6.6 TESTING A SINGLE BINOMIAL PROPORTION

The binomial distribution depends on two parameters n and p. It rep-
resents the sum of n independent Bernoulli trials. A Bernoulli trial is a
test with two possible outcomes that are often labeled as success and
failures. The binomial random variable is the total number of successes
out of the n trials. So the binomial random variable can take on any
value between 0 and n. The binomial distribution has mean equal to
np and variance np (1 − p ). These results are to construct the pivotal