Answer to Question #128228 in Statistics and Probability for mihir

Question

The mean height of 1000 students at a certain college is 165 cms and S.D  is  10 cms. Assuming  normal  distribution , find  the  number  of students whose height is a).Greater than 172 cm. b).between 159 and 178cm.

Answer 1

a) P(x > 172 cm)

Let the population mean be $\mu$ = 165.

Let the population standard deviation be $\delta$ = 10.

To get the P(x > 172 cm) we use the formula z = (x-μ)/σ;

z = (172 - 165)/10 = 0.7.

P(x > 172 cm) = P(z > 0.7) = 1 - P(z < 0.7)

$\implies$ 1 - 0.75804

= 0.24196

To obtain the number of students whose height is expected to be more than 172 cm is obtained using the binomial formula

E(X) = np; where n = 1000 and p = 0.24196;

= 1000 * 0.24196

= 241.96

=242

b) P(159 < x < 178 cm)

At the height of up to 159

P(x < 159) we use the following formula z = (x-μ)/σ;

z = (159-165)/10 = -0.6;

P(x < 159) = P(z < -0.6) = 0.27425.

At the height of up to 178

P(x < 178) we use the following formula z = (x-μ)/σ;

z = (178 - 165)/10 = 1.3;

P(x < 178) = P(z < 1.3) = 0.90320; 

P(159 < x < 178 cm) = P(-0.6 < z < 1.3)

= 0.90320 - 0.27425

= 0.62895.

To get the number of students whose height is expected to be between 159 cm and 178 cm out of 1000 students, we find the expectation of a binomial distribution.

E(X) = np ; where n= 1000 and p = 0.62895;

= 1000 * 0.62895

= 628.95

= 629