Question

NotationListed below are body temperatures from five different\({\bf{\bar d}}\)subjects measured at 8 AM and again at 12 AM (from Data Set 3 “Body Temperatures” in Appendix B). Find the values of and\({{\bf{s}}_{\bf{d}}}\). In general, what does\({{\bf{\mu }}_{\bf{d}}}\)represent?

Temperature\(\left( {^{\bf{0}}{\bf{F}}} \right)\)at 8AM	97.8	99.0	97.4	97.4	97.5
Temperature\(\left( {^{\bf{0}}{\bf{F}}} \right)\)at 12AM	98.6	99.5	97.5	97.3	97.6

Short answer

The value of\(\bar d\)is equal to -0.28.

The value of\({s_d}\)is equal to 0.36.

\({\mu _d}\) is the mean value of the differences for the population of all matched pairs of temperatures at 8 AM and 12 AM.

Step-by-step solution

Given information

The body temperature of a set of five subjects is recorded, first at 8 AM and then at 12 AM. Therefore, the sample size is equal to \(n = 5\).

Differences of matched pairs

The following table shows the values of the differences for each pair by subtracting the temperature at 12 AM from the temperature at 8 AM.

Temperature\(\left( {^0F} \right)\)at 8 AM	97.8	99.0	97.4	97.4	97.5
Temperature\(\left( {^0F} \right)\)at 12 AM	98.6	99.5	97.5	97.3	97.6
Differences\(\left( {{d_i}} \right)\)	-0.8	-0.5	-0.1	0.1	-0.1	\(\sum {{d_i}} = - 1.4\)

Mean value of the differences

Themean value of the differences between the body temperature at 8 AM and 12 AM is equal to:

\(\begin{array}{c}\bar d = \frac{{\sum d }}{n}\\ = \frac{{ - 1.4}}{5}\\ = - 0.28\end{array}\)

Therefore, the mean value \(\left( {\bar d} \right)\)is -0.28.

Compute the standard deviation

Thestandard deviation of the differences between the body temperatures at 8 AM and 12 AM is equal to:

\(\begin{array}{c}{s_d} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{\left( {{d_i} - \bar d} \right)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( { - 0.8 - \left( { - 0.28} \right)} \right)}^2} + {{\left( { - 0.5 - \left( { - 0.28} \right)} \right)}^2} + ...... + {{\left( { - 0.1 - \left( { - 0.28} \right)} \right)}^2}}}{{5 - 1}}} \\ = 0.36\end{array}\)

Therefore, the standard deviation \(\left( {{s_d}} \right)\)is equal to 0.36.

Representation of the mean value

In general,\({\mu _d}\)represents the mean value of all the differences computed for the pairs of data when the population of all matched pairs of temperatures at 8 AM and 12 AM is taken into consideration.