Question

Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes. a) What is the mean of the time interval? (2 points) b) What is the probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 minutes? (6 points) c) What is the standard deviation of the time interval? (2 points) d) W hat is the probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes? (4 points)

Answer 1

Answer:

a) The mean of the time interval is 45 minutes.

b) 27.78% probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 minutes

c) The standard deviation of the time interval is 25.98 minutes.

d) 11.11% probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes

Step-by-step explanation:

An uniform probability is a case of probability in which each outcome is equally as likely.

For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.

The probability that we find a value X lower than x is given by the following formula.

$P(X < x) = \frac{x - a}{b-a}$

The probability of finding a value X between c and d is:

$P(c \leq X \leq d) = \frac{d-c}{b-a}$

The mean of the uniform distribution is:

$M = \frac{a+b}{2}$

The standard deviation of the uniform distribution is:

$S = \sqrt{\frac{(b-a)^{2}}{12}}$

Uniform distribution over an interval from 0 to 90 minutes.

This means that $a = 0, b = 90$

a) What is the mean of the time interval?

$M = \frac{a+b}{2} = \frac{0 + 90}{2} = 45$

The mean of the time interval is 45 minutes.

b) What is the probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 minutes?

$P(10 \leq X \leq 35) = \frac{35-10}{90-0} = 0.2778$

27.78% probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 minutes.

c) What is the standard deviation of the time interval?

$S = \sqrt{\frac{(90-0)^{2}}{12}} = 25.98$

The standard deviation of the time interval is 25.98 minutes.

d) What is the probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes?

$P(X < 10) = \frac{10 - 0}{90 - 0} = 0.1111$

11.11% probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes