The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 3.6 cars and a standard deviation of 5 . The number of cars running the red light was observed on 100 randomly chosen days and the mean number of cars calculated. Describe the sampling distribution of the sample mean.

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".

Solution to the problem

Let X the random variable that represent the number of cars running a red light of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(3.6,5)[/tex]

Where [tex]\mu=3.6[/tex] and [tex]\sigma=5[/tex]

Since the distribution for X is normal then 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]

With the following parameters:

[tex]\mu_{\bar X}= 3.6[/tex]

[tex]\sigma_{\bar X} = \frac{5}{\sqrt{100}}= 0.5[/tex]