Question

The weights of people in a certain population are normally distributed with a mean of 159 l b and a standard deviation of 25 l b. For samples of size 7​, determine whether the distribution of x overbar is normal or approximately normal and give its mean and standard deviation.

Answer 1

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

`X \sim N(159,25)`

Where
`\mu=159` and
`\sigma=25`

Since the distribution of X is normal than we can conclude that we know the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

With the following parameters:

`\mu_(\bar X) = 159`

`\sigma_(\bar X) =(25)/(√(7)) = 9.449`

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

`X \sim N(159,25)`

Where
`\mu=159` and
`\sigma=25`

Since the distribution of X is normal than we can conclude that we know the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

With the following parameters:

`\mu_(\bar X) = 159`

`\sigma_(\bar X) =(25)/(√(7)) = 9.449`