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# System prepended metadata

title: PID-638

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For the given data, we are asked to find the mean, variance, and standard deviation of the sampling distribution of the sample mean for two different sample sizes, $n=1$ and $n=2$. 

For a sample of size $n$ drawn from a population with mean μ and standard deviation $σ$, the mean of the sampling distribution of the sample mean is $μ$ and the standard deviation of the sampling distribution of the sample mean is $σ/√n.$ 

For the given data, the population mean is:

$$μ = (2+5+1+4+7)/5 = 3.8$$

The population variance is:

$$σ^2 = [(2-3.8)^2 + (5-3.8)^2 + (1-3.8)^2 + (4-3.8)^2 + (7-3.8)^2]/5 = 4.56.$$


The population standard deviation is:

$$σ = \sqrt{4.56} = 2.13542.$$


For $n=1$:

a. The mean of the sampling distribution of the sample mean is μ = 3.8.

b. The variance of the sampling distribution of the sample mean is 
$$σ^2/n = 4.56/1 = 4.56.$$

c. The standard deviation of the sampling distribution of the sample mean is $$\frac{σ}{√n} = 2.13542/1 = 2.13542.$$

d. Since $n=1$, the sampling distribution of the sample mean is the same as the population distribution. Therefore, the histogram of the sampling distribution of the sample mean is the same as the histogram of the population distribution.

For $n=2:$

a. The mean of the sampling distribution of the sample mean is $$μ = 3.8.$$
b. The variance of the sampling distribution of the sample mean is $$σ^2/n = 4.56/2 = 2.28.$$
c. The standard deviation of the sampling distribution of the sample mean is $$σ/√n = 2.13542/√2 = 1, 50997.$$
d. To draw a histogram of the sampling distribution of the sample mean, we need to take all possible samples of size $2$ from the population and calculate the mean of each sample. Then, we can plot a histogram of these sample means. However, since the population size is small, we can simply calculate the mean of all possible samples of size 2. There are $10$ possible samples of size $2$ from the population: $(2,5), (2,1), (2,4), (2,7), (5,1), (5,4), (5,7), (1,4), (1,7),$ and $(4,7)$. The mean of each of these samples is:

$(2+5)/2 = 3.5$
$(2+1)/2 = 1.5$
$(2+4)/2 = 3$
$(2+7)/2 = 4.5$
$(5+1)/2 = 3$
$(5+4)/2 = 4.5$
$(5+7)/2 = 6$
$(1+4)/2 = 2.5$
$(1+7)/2 = 4$
$(4+7)/2 = 5.5$

We can then plot a histogram of these sample means. Here's an example of what the histogram might look like:

![Histogram of sample means for n=2](https://mse.redwoods.edu/darnold/math15/spring2013/R/Activities/DistributionOfSampleMeans_files/figure-html/unnamed-chunk-4-1.png) [2]