Question

If the quality of individual boards varies according to a normal distribution with mean µ = 100 and standard deviation σ = 4 , what will be the distribution of the sample averages? (Recall the sample size is n = 5.)

Answer 1

The answer is, with a mean 100 and a standard deviation of 4/ root 5.

Explanation:

From the given question, let us recall the following,

Normal distribution with µ = 100

The standard deviation σ =4

the sample size n =5

Therefore,

The distribution of x will be normal with the mean 100 and a standard deviation of 4/ root 5,

Which is denoted as,

Mean 100 = 4√5

Answer 2

The distribution of sample averages will have a normal distribution withe mean = 100 and a stamdard deviation of 1.789.

Explanation:

According to the central limit theorem, the mean of the distribution of sample means is equal to the population mean.

μₓ = μ = 100

The theorem goes further to explain that if samples of finite size are obtained from the population, the distribution of sample means will approximate the distribution of the population. Since the population is known to have data that is distributed normally, the distribution of sample means will have a normal distribution too.

The standard deviation of the distribution of sample means is given as the standard error of the mean,

σₓ = (σ/√n)

where σ = population standard deviation = 4

n = sample size = 5

σₓ = (4/√5) = 1.789

Hope this Helps!!!