Answer to Question #175188 in Statistics and Probability for Ella

Question

The mean weight of 500 college students is 70 kg and the standard deviation is 5 kg

Assuming that the weight is normally distributed, determine how many students weights:

a: Between 60 kg and 75 kg

b. More Than 90 kg

c. less than 65 kg

Answer 1

By condition, $a = 70\,\,\sigma = 5$

a: Between 60 kg and 75 kg

Find the probability that the student's weight will be between 60 and 75 kg:

$P(60 < x < 75) = \Phi \left( {\frac{{\beta - a}}{\sigma }} \right) - \Phi \left( {\frac{{\alpha - a}}{\sigma }} \right) = \Phi \left( {\frac{{75 - 70}}{5}} \right) - \Phi \left( {\frac{{60 - 70}}{5}} \right) = \Phi \left( 1 \right) + \Phi \left( 2 \right) = 0,3413 + 0,4772 = {\rm{0}}{\rm{,8185}}$

Then quantity of students weights between 60 kg and 75 kg is

${\rm{500}} \cdot {\rm{0}}{\rm{,8185}} \approx {\rm{409}}$

Answer: 409

b. More Than 90 kg

$P(x > 90) = \Phi \left( \infty \right) - \Phi \left( {\frac{{90 - 70}}{5}} \right) = \Phi \left( \infty \right) - \Phi \left( 4 \right)\approx 0,5 - 0,5 \approx 0$

Then quantity of students weights more then 90 kg is 0.

Answer: 0

c. less than 65 kg

$P(x < 65) = P(0 < x < 65) = \Phi \left( {\frac{{65 - 70}}{5}} \right) - \Phi \left( {\frac{{0 - 70}}{5}} \right) = \Phi \left( {14} \right) - \Phi \left( 1 \right) = 0,5 - 0,3413= {\rm{0}}{\rm{,1587}}$

Then quantity of students weights less then 65 kg is ${\rm{500}} \cdot {\rm{0}}{\rm{,1587}} \approx 79$

Answer: 79