Question

Males in the Netherlands are the tallest, on average, in the world with an average height of 183 centimeters (cm) (BBC News website). Assume that the height of men in the Netherlands is normally distributed with a mean of 183 cm and standard deviation of 10.5 cm.

a.What is the probability that a Dutch male is shorter than 175 cm?

b.What is the probability that a Dutch male is taller than 195 cm?

c.What is the probability that a Dutch male is between 173 and 193 cm?

d.Out of a random sample of 1000 Dutch men, how many would we expect to be taller than 190 cm?

Answer 1

Solution :

Given that ,

mean = = 183

standard deviation = = 10.5

a)

P(x < 175) = P((x - ) / < (175 - 183 ) / 10.5)

= P(z < -0.76)

= 0.2236 Using standard normal table,

The probability that a Dutch male is shorter than 175 cm is 0.2236

b)

P(x > 195) = 1 - P(x < 195)

= 1 - P((x - ) / < (195 - 183) / 10.5)

= 1 - P(z < 1.14)

= 1 - 0.8729 Using standard normal table.

= 0.1271

The probability that a Dutch male is taller than 195 cm is 0.1271

c)

P(173 < x < 193) = P((173 - 183)/ 10.5) < (x - ) /  < (193 - 183) / 10.5) )

= P(-0.95 < z < 0.95)

= P(z < 0.95) - P(z < -0.95)

= 0.8289 - 0.1711

= 0.6578

The probability that a Dutch male is between 173 and 193 cm is 0.6578

d)

P(x > 190) = 1 - P(x < 190)

= 1 - P((x - ) / < (190 - 183) / 10.5)

= 1 - P(z < 0.67)

= 1 - 0.7486 Using standard normal table.

= 0.2514

Probability = 0.2514

Out of a random sample of 1000 Dutch men,

= 0.2514 * 1000

= 251.4

= 252

252 would we expect to be taller than 190.