Dead battery? a car company has found that the lifetime of its batteries varies from car to car according to a normal distribution with mean μ = 48 months and standard deviation σ = 8.2 months. the company installs a new brand of battery on an srs of 8 cars.
(a) if the new brand has the same lifetime distribution as the previous type of battery, describe the sampling distribution of the mean lifetime .
(b) the average life of the batteries on these 8 cars turns out to be = 42.2 months. find the probability that the sample mean lifetime is 42.2 months or less if the lifetime distribution is unchanged. what conclusion would you draw?

a)

b)

Step-by-step explanation:

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Part a

From the central limit theorem we know that the distribution for the sample mean is given by:

Part b

We want this probability

In order to answer this question we can use the z score in order to find the probabilities, the formula given by:

And using a calculator, excel or the normal standard table we have that:

step-by-step explanation:

the answer is 128