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

Required:
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?

Respuesta :

Complete Question

Height of Dutch Men. 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:

a. 0.2231

b. 0.1266

c. 0.6591

d. 252 people

Step-by-step explanation:

We solve using the z score formula of

z = (x-μ)/σ, where

x is the raw score

μ is the population mean = 183 cm

σ is the population standard deviation = 10.5 cm

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

This is interpreted as when x < 175 cm

Hence:

z = 175 - 183/10.5

z = -0.7619

Probability value from Z-Table:

P(x<175) = 0.22306

Approximately ≈ 0.2231

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

This is interpreted as when x > 195 cm

Hence:

z = 195 - 183/10.5

z = 1.14286

Probability value from Z-Table:

P(x<195) = 0.87345

P(x>195) = 1 - P(x<195) = 0.12655

Approximately ≈ 0.1266

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

For x = 173

z = 173 - 183/10.5

z = -0.95238

Probability value from Z-Table:

P(x = 173) = 0.17045

For x = 193

Hence:

z = 193 - 183/10.5

z = 0.95238

Probability value from Z-Table:

P(x= 193) = 0.82955

Hence, the probability that a Dutch male is between 173 and 193 cm

P(x = 193) - P(x = 173)

0.82955 - 0.17045

= 0.6591

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

This is interpreted as when x > 190 cm

Hence,

z = 190 - 183/10.5

= 0.66667

Probability value from Z-Table:

P(x<190) = 0.74751

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

= 0.25249

Therefore, the number of people we would expect to be taller than 190 cm is

Random number of samples × Probability

= 1000 × 0.25249

= 252.49 people

Approximately≈ 252 people

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