Question

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 2.4 cars and a standard deviation of 4. 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.

Answer 1

The sampling distribution of the sample mean is:

`\bar X\sim N(\mu_(\bar x)=2.4,\ \sigma_(\bar x)=0.40)`

Explanation:

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Then, the mean of the distribution of sample means is given by,

`\mu_(\bar x)=\mu`

And the standard deviation of the distribution of sample means is given by,

`\sigma_(\bar x)=(\sigma)/(√(n))`

Let X = number of cars running a red light in a day, at a given intersection.

The information provided is:

`E(X)=\mu=2.4\\SD(X)=\sigma=4\\n=100`

The sample selected is quite large, i.e. n = 100 > 30.

The Central limit theorem can be used to approximate the sampling distribution of the sample mean number of cars running a red light in a day, by the Normal distribution.

The mean of the sampling distribution of the sample mean is:

`\mu_(\bar x)=\mu=2.4`

The standard deviation of the sampling distribution of the sample mean is:

`\sigma_(\bar x)=(\sigma)/(√(n))=(4)/(√(100))=0.40`

The sampling distribution of the sample mean is:

`\bar X\sim N(\mu_(\bar x)=2.4,\ \sigma_(\bar x)=0.40)`