The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Twenty-five gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 25 gas stations. What is the distribution to use for the average cost of gasoline for the 25 gas stations?

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".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

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".

Let X the random variable that represent the cost of unleaded gasoline in the Bay Area of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(4.59,0.1)[/tex]

Where [tex]\mu=4.59[/tex] and [tex]\sigma=0.1[/tex]

2) Solution to the problem

And let [tex]\bar X[/tex] represent the sample mean, by the central limit theorem, the distribution for the sample mean is given by:

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

On this case if we replace we have:

[tex]\bar X \sim N(65,\frac{0.1}{\sqrt{25}}=0.02)[/tex]

So we have the following parameters:

[tex]\mu_{\bar X}=65[/tex]

[tex]\sigma_{\bar X}=0.02[/tex]