Question

Tire lifetime:the lifetime of a certain type of automobile tire(in thousands of moles) is normally distributed with mean u=41 and standard deviation o=4 What is the probability that a radon chosen tire has lifetime greater than 50 thousand miles? What proportion of tires have lifetime between 36 and 45 thousand miles What proportion of tires have lifetimes less than 46 thousand miles? Round answers at least 4 places

Answer 1

Let X: Lifetime of a automobile tire have normal distribution with mean and standard deviation .

a) Probability that a random chosen tire has lifetime greater than 50 that is P(X > 50)

Covert the given X into z score, the formula of z score is

The z score for x = 50 is,

That is P(X > 50) becomes P(Z > 2.25)

The probability using z table for z = 2.25 is 0.9878.

But the table provides the less then probability, just subtract less then from 1 to get greater than probability.

1 - 0.9878 = 0.0122

Therefore, the probability that a random chosen tire has lifetime greater than 50 thousands miles is 0.0122

b) Proportion of tire have lifetime between 36 and 45 thousand miles that is P(36 < X < 45).

convert both x into z scores,

The z score for x = 36 is,

and the z score for x = 45 is,

That is P(36 < X < 45) becomes P(-1.25 < z < 1)

The probability for z = -1.25 is 0.1057 and the probability for z = 1 is 0.8413

The between probability is 0.8413 - 0.1057 = 0.7356

Therefore, the proportion of tire have lifetime between 36 and 45 thousand miles is 0.7356

c) Proportion of tires have lifetimes less than 46 thousands miles that is P(X < 46)

Convert x = 46 into z score

That is, P(X < 46) becomes P(z < 1.25)

The probability for z = 1.25 using the z table is 0.8944

Therefore, the proportion of tires have lifetime less than 46 thousands of miles is 0.8944