Question

A manufacturer of auto engine oil claims that cars using their product can go an average of 180 days without changing oil, with a standard deviation of 4 days. The distribution of the length of time between oil changes with this engine oil is known to be left-skewed. To test this claim a gas station owner takes a sample of 100 cars using this brand, and computes the sample mean. Assuming that the manufacturer’s claim is correct, which of the following describes the sampling distribution of the sample mean?

Select one:

a. An approximately normal distribution with mean 180 and standard deviation of 4.

b. A distribution that is approximately standard normal.

c. A right-skewed distribution with mean 180 and standard deviation of 0.4.

d. A distribution with mean 180 and standard deviation of 0.4 but with unknown shape because the shape of the population is right-skewed.  

e. An approximately normal distribution with mean 180 and variance of 0. 4.

Answer 1

According to the central limit theorem any population with given mean and variance and sample size > 30 will have normal distribution.

Hence the distribution is normal.

Now, the parameters,

Mean = 180 days

standard deviation = = 4 days

Sample size = n = 100

Now, for sample

Sample mean = Mean = 180 days

Sample standard deviation =

Sample standard deviation =

Sample standard deviation =

Sample standard deviation = 0.4

The correct answer is:

a. An approximately normal distribution with mean 180 and standard deviation of 4.