Question

Question 4: The air pressure in a randomly selected tire of a new car of a certain model is normally distributed with a mean value 32 psi and standard deviation of 1.5 psi.

What is the probability that the pressure in a randomly selected tire is between 30 psi and 32 psi?

Calculate the IQR of the distribution of tire pressure of all cars of this model?

What is minimum tire pressure for the top 5% of the cars?

Answer 1

4)

Solution :

Given that ,

mean = = 32

standard deviation = = 1.5

1)

P(30 < x < 32) = P((30 - 32 / 1.5) < (x - ) / < ( 32 - 32) / 1.5) )

= P(-1.33 < z < 0)

= P(z < 0) - P(z < -1.33)

= 0.5 - 0.0918 = 0.4082

Probability = 0.4082

2) IQR

P(Z < z) = 0.25

P(Z < -0.2533) = 0.25

z = -0.2533

Using z-score formula,

x = z * +

x = -0.2533 * 1.5 + 32 = 31.62 and

Q₁ = 31.62

P(Z < z) = 0.75

P(Z < 0.2533) = 0.75

z = 0.2533

Using z-score formula,

x = z * +

x = 0.2533 * 1.5 + 32 = 32.38

Q₃ = 32.38

IQR = Q₃ - Q₁ = 32.38 - 31.62 = 0.76

3)

P(Z > z) = 5%

1 - P(Z < z) = 0.05

P(Z < z) = 1 - 0.05 = 0.95

P(Z < 1.645) = 0.95

z = 1.645

Using z-score formula,

x = z * +

x = 1.645 * 1.5 + 32 = 34.4675 = 34.47

Minimum tyre pressure = 34.47