Question

The body temperatures of a group of healthy adults have a​ bell-shaped distribution with a mean of 98.08degreesF and a standard deviation of 0.63degreesF. Using the empirical​ rule, find each approximate percentage below. a. What is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the​ mean, or between 97.45degreesF and 98.71degrees​F? b. What is the approximate percentage of healthy adults with body temperatures between 96.19degreesF and 99.97degrees​F?

Answer 1

Solution :

Given that,

mean = = 98.08

standard deviation = = 0.63

a) P(97.45 < x < 98.71) = P[(97.45 - 98.08)/ 0.63) < (x - ) /  < (98.71 - 98.08) / 0.63 ) ]

= P(-1.00 < z < 1.00)

= P(z < 1.00) - P(z < -1.00)

Using Empirical rule,

  P( - < x <   + ) = 68%

  P( 98.08 - 0.63 < x < 98.08 + 0.63) = 68%

P(97.45 < x < 98.71) = 68%

b) P(96.19 < x < 99.97) = P[(96.19 - 98.08)/ 0.63) < (x - ) /  < (99.97 - 98.08) / 0.63 ) ]

= P(-3.00 < z < 3.00)

= P(z < 3.00) - P(z < -3.00)

Using Empirical rule,

  P( - 3 < x <   + 3 ) = 99.7%

  P( 98.08 - 3 * 0.63 < x < 98.08 + 3 * 0.63) = 99.7%

  P( 98.08 - 1.89 < x < 98.08 + 1.89) = 99.7%

P(96.19 < x < 99.97) = 99.7%