Download Mean Body Temperature

Mean Body Temperature
Stat 305 Spring Semester 2006
Most people when asked will identify 98.6 F as the mean body temperature for healthy
adults. The table below list the body temperatures of 106 such adults. (This is an actual
data set obtained by University of Maryland researchers.)
98.6
98.8
98.5
99.6
98.0
96.9
98.8
98.6
98.9
98.0
Body Temperature (F ) of
98.6 98.0 98.0 99.0 98.4
98.6 97.0 97.0 98.8 97.6
97.3 98.7 97.4 98.9 98.6
98.7 99.4 98.2 98.0 98.6
97.8 98.0 98.4 98.6 98.6
97.6 97.1 97.9 98.4 97.3
98.7 97.8 98.0 97.1 97.4
98.3 98.7 98.8 99.1 98.6
98.4 98.6 97.1 97.9 98.8
98.4 97.8 98.4 97.4 98.0
106 Healthy
98.4 98.4
97.7 98.8
99.5 97.5
98.6 97.2
97.8 99.0
98.0 97.5
99.4 98.4
97.9 98.8
98.7 97.6
97.0
Adults
98.4
98.0
97.3
98.4
96.5
97.6
98.6
98.0
98.2
98.6
98.0
97.6
98.6
97.6
98.2
98.4
98.7
99.2
98.6
98.3
98.2
98.2
98.0
98.5
98.5
98.5
97.8
The data set can be viewed as a random sample fX1 ; X2 ; : : : ; X106 g drawn from the distribution of the random variable X denoting the body temperature. The sample mean is
X n = 98:20 F for the data set in the table. Previous studies suggest that the standard
deviation of X is = 0:62 F. (In practice, the value of has to be estimated. For the
moment, we will assume that the standard deviation of X is known and is equal to 0:62 F.
In fact, the value given, = 0:62 F, is the sample standard deviation of the data set.)
Based on the sample data, the key question is the following. Should the conventionally
held belief that the mean body temperature of healthy adults is 98.6 F be discarded or should
we maintain this belief? We should abandon 98.6 F as the mean body temperature only
if there is ‘enough’ evidence (whatever that means) to support such a move. Certainly,
we would want to give the conventional temperature the bene…t of the doubt, not unlike
a person accused of a crime and standing trial. (Such a person is assumed innocent until
proven guilty.) Therefore, the probability of rejecting 98.6 F as the mean body temperature
when it is indeed the mean body temperature should be small, but how small?
There is some evidence that 98.6 F should be abandoned as the mean body temperature.
Since the sample size is large, by the Central Limit Theorem,p we can assume that X n is
approximately normal with mean and standard deviation = 106. Using the data in the
table, a 95% con…dence interval for is
p
p
(X n z:025 ( = 106); X n + z:025 ( = 106)) = (98:08; 98:32).
Notice the value 98.6 is not in this interval!
In the …nal analysis, we have to decide between the following two competing alternatives
or hypotheses.
Null hypothesis (H0 ):
= 98.6
Alternative hypothesis (H1 ):
6= 98.6
(represents the status quo)
(represents a rejection of the status quo)
(In some texts, the alternative hypothesis is denoted as (Ha ).
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