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 57 inches?

2. What is the probability that a randomly chosen 10 year old is between 61 and 63 inches?

3. If the shortest 15% 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 24 th percentile?  inches

Answer 1

Solution :

Given that ,

mean = = 55

standard deviation = = 6

1)

P(x < 57) = P((x - ) / < (57 - 55) / 6)

= P(z < 0.33)

Using standard normal table,

P(x < 57) = 0.6293

Probability = 0.6293 = 0.63

2)

P(61 < x < 63) = P((61 - 55)/ 6) < (x - ) / < (63 - 55) / 6) )

= P(1 < z < 1.33)

= P(z < 1.33) - P(z < 1)

= 0.9082 - 0.8413 = 0.0669

Probability = 0.07

3)

P(Z < z) = 15% = 0.15

P(Z < -1.04) = 0.15

z = -1.04

Using z-score formula,

x = z * +

x = -1.04 * 6 + 55 = 48.76

Cutoff = 48.76

4)

P(Z < z) = 24% = 0.24

P(Z < -1.04) = 0.24

z = -0.71

Using z-score formula,

x = z * +

x = -0.71 * 6 + 55 = 50.74

24th percentile = 50.74 inches