Question

# 5. Horses in a stable have a mean weight of 950 pounds with a standard deviation...

5. Horses in a stable have a mean weight of 950 pounds with a standard deviation of 77 pounds. Weights of horses follow the normal distribution. One horse is selected at random.

a) What is the probability that the horse weighs less than 900 pounds?

b) What is the probability that the horse weigh more than 1,100 pounds?

c) What is the probability that the horse weighs between 900 and 1,100 pounds?

d) What weight is the 90th percentile? (Round to the nearest pound)

Solution :

Given that ,

a)

P(x < 900) = P[(x - ) / < (900 - 950) / 77]

= P(z < -0.65)

= 0.2578

Probability = 0.2578

b)

P(x > 1100) = 1 - P(x < 1100)

= 1 - P[(x - ) / < (1100 - 950) / 77)

= 1 - P(z < 1.95)

= 1 - 0.9744

= 0.0256

Probability = 0.0256

c)

P(900 < x < 1100) = P[(900 - 950)/ 77) < (x - ) /  < (1100 - 950) / 77) ]

= P(-0.65 < z < 1.95)

= P(z < 1.95) - P(z < -0.65)

= 0.9744 - 0.2578

= 0.7166

Probability = 0.7166

d)

Using standard normal table ,

P(Z < z) = 90%

P(Z < 1.28) = 0.9

z = 1.28

Using z-score formula,

x = z * +

x = 1.28 * 77 + 950 = 1049

90th percentile is 1049 pound