Concerns about climate change and CO2 reduction have initiated the commercial production of blends of biodiesel (e.g., from renewable sources) and petrodiesel (from fossil fuel). Random blended fuel samples of size 35 are tested in a lab to ascertain the bio/total carbon ratio (X). If the true (i.e. population) mean is 0.948 and the true (i.e. population) standard deviation is 0.006, what is the distribution of Xbar?

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".

Let X the random variable who represents the bio/total carbon ratio. We know from the problem that the distribution for the parameters for the random variable X are:

[tex]\mu = 0.948[/tex]

[tex]\sigma=0.006[/tex]

We select a sample of n=35 nails. That represent the sample size.

From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

So on this case :

[tex]\bar X \sim N(\mu=0.948, \sigma_{\bar X}=\frac{0.006}{\sqrt{35}}=0.00101)[/tex]