Question

For the population of one town, the number of siblings, x, is a random variable whose relative frequency histogram has a reverse J-shape. The mean number of siblings is 1.3 and the standard deviation is 1.4. Let denote the mean number of siblings for a random sample of size 30. Determine the sampling distribution of the mean for samples of size 30, and give its mean and standard deviation

Answer 1

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

The mean is given by:

`\mu_(\bar X)= 1.3`

And the deviation is given by:

`\sigma_(\bar X) =(\sigma)/(√(n))= (1.4)/(√(30))= 0.256`

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

`X \sim N(1.3,1.4)`

Where
`\mu=1.3` and
`\sigma=1.4`

The sample mean is given by this formula:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

The sample size is n =30, and the distribution for the sample mean is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

The mean is given by:

`\mu_(\bar X)= 1.3`

And the deviation is given by:

`\sigma_(\bar X) =(\sigma)/(√(n))= (1.4)/(√(30))= 0.256`