Question

In the 19th century, it was determined by experimentation that human body temperature is normally distributed N(98.6, 0.6) (in Fahrenheit F). Recently a new study has proposed that this should be modified to indicate that human body temperature is approximately normally distributed N(98.2,0.7) (F).

a. Which of these two have a larger range of temperatures that represent 95% of the most common body temperatures? Provide an analysis that supports your conclusion.

b. Over the course of 20 years an individual notices that their average body temperature is 97.6 degrees F. What percentile do they fall into in each of the scales above?

Answer 1

a) P(X < x) = 0.025

or, P((X - )/ < (x - )/ = 0.025

or, P(Z < (x - 98.6)/0.6) = 0.025

or, (x - 98.6)/0.6 = -1.96

or, x = -1.96 * 0.6 + 98.6

or, x = 97.424

P(X > x) = 0.025

or, P((X - )/ > (x - )/ = 0.025

or, P(Z > (x - 98.6)/0.6) = 0.025

or, P(Z < (x - 98.6)/0.6) = 0.975

or, (x - 98.6)/0.6 = 1.96

or, x = 1.96 * 0.6 + 98.6

or, x = 99.776

For 19th century, the range of temperatures that represent 95% of the most common body temperatures is from 97.424 to 99.776.

P(X < x) = 0.025

or, P((X - )/ < (x - )/ = 0.025

or, P(Z < (x - 98.2)/0.7) = 0.025

or, (x - 98.2)/0.7 = -1.96

or, x = -1.96 * 0.7 + 98.2

or, x = 96.828

P(X > x) = 0.025

or, P((X - )/ > (x - )/ = 0.025

or, P(Z > (x - 98.2)/0.7) = 0.025

or, P(Z < (x - 98.2)/0.7) = 0.975

or, (x - 98.2)/0.7 = 1.96

or, x = 1.96 * 0.7 + 98.2

or, x = 99.572

For the new study, the range of temperatures that represent 95% of the most common body temperatures is from 96.828 to 99.572.

So the new study has a larger range of temperatures that represent 95% of the most common body temperatures.

b) For the 19th Century

P(X < 97.6)

= P((X - )/ < (97.6 - )/)

= P(Z < (97.6 - 98.6)/0.6)

= P(Z < -1.67)

= 0.0475 = 4.75%

For the new study

P(X < 97.6)

= P((X - )/ < (97.6 - )/)

= P(Z < (97.6 - 98.2)/0.7)

= P(Z < -0.86)

= 0.1949 = 19.49%