Question

Heights of 10 year olds. Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. Round all answers to two decimal places.

1. What is the probability that a randomly chosen 10 year old is shorter than 48 inches?

2. What is the probability that a randomly chosen 10 year old is between 50 and 51 inches?

3. If the shortest 10% of the class is considered very tall, what is the height cutoff for very tall? inches

4. What is the height of a 10 year old who is at the 34 th percentile? inches

Answer 1

Solution :

Given that ,

mean = = 55

standard deviation = = 6

1)

P(x < 48) = P((x - ) / < (48 - 55) / 6)

= P(z < -1.17)

Using standard normal table,

P(x < 48) = 0.121

Probability = 0.121

2)

P(50 < x < 51) = P((50 - 55)/ 6) < (x - ) / < (51 - 55) / 6) )

= P(-0.83 < z < -0.67)

= P(z < -0.67) - P(z < -0.83)

= 0.2514 - 0.2033 = 0.0481

Probability = 0.0481

3 )

P(Z < z) = 10% = 0.10

P(Z < -1.28) = 0.10

z = -1.28

Using z-score formula,

x = z * +

x = -1.28 * 6 + 55 = 47.32

Height cutoff = 47.32

4)

P(Z < z) = 34%

P(Z < -0.41) = 0.34

z = -0.41

Using z-score formula,

x = z * +

x = -0.41 * 6 + 55 = 52.54

34th percentile = 52.54