Question

Suppose that it is known that the time it takes a certain light bulb to burn out is normally distributed with mean time 1450 hours and standard deviation 145 hours.

a.Describe the shape of the sampling distribution of ?̅if 10 light bulbs are randomly selected. Justify your answer.

b.Find the mean and standard deviation of the sampling distribution of ?̅if 10 light bulbs are randomly selected.

c.Find the proportion of all light bulbs produced that take more than 1500 hours to burn out.

d.The company advertises that the light bulbs last at least 1350 hours. A government agency will test their light bulbs to ensure that the light bulbs last at least as long as advertised. If the agency finds that the light bulbs last less than the advertised time, the company can face fines. If the government agency will take a sample of 10 randomly selected light bulbs and compute the mean of the time it takes the light bulbs to burn out to determine how long the light bulbs last, what is the probability that the company will be fined?

Answer 1

a. The shape of the sampling diatribution will follow a normal curve (bell-shaped curve). Because, the samples are selected from the population follows a normal distribution.

b. Mean and standard distribution,

The sample mean will be the same as the population mean,

The sampling standard deviation is given as,

c. proportion of all light bulbs produced that take more than 1500 hours to burn out is,

We know that,

z = (1500- 1450)/145

z = 0.3448

P(z > 0.3448) = 1- p(z < 0.3448)

= 0.3651

= 36.51%

d.

Probability of mean being less than1350hours for a randomly selected 10 samples is,

z = - 0.7027

P(z < - 0.7027) = 0.2411

= 24.11%