Question

A random sample of 81 automobiles traveling on a section of an interstate showed an average speed of 60 mph. The distribution of speeds of all cars on this section of highway is normally distributed, with a standard deviation of 13.5 mph.

The value to use for the standard error of the mean is:

1.13.5
2.9
3.2.26
4.1.5

Answer 1

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

`X \sim N(60,13.5)`

Where
`\mu=60` and
`\sigma=13.5`

And for this case we select a sample size of n= 81. Since the distribution for X is normal then we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And the standard error of the mean would be:

`\sigma_(\bar X) =(13.5)/(√(81))= 1.5`

4.1.5

Explanation:

Previous concepts

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 variable of interest of a population, and for this case we know the distribution for X is given by:

`X \sim N(60,13.5)`

Where
`\mu=60` and
`\sigma=13.5`

And for this case we select a sample size of n= 81. Since the distribution for X is normal then we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And the standard error of the mean would be:

`\sigma_(\bar X) =(13.5)/(√(81))= 1.5`

4.1.5