Week11 practice 2

Sampling and statistics

We will work in pairs.

Generators: We shall perform sampling from two random distribution the normal distribution and a uniform distribution over some range $[a,b]$.

For a normal distribution, choose the mean $\mu$ and its s.d. $\sigma$. For a uniform distribution, choose a and b. (Don't forget to take note of the actual distribution that you use so that you can tell your statisticians the correct answers.)

Generate 10 sets of data:

• For normal distributions (with different parameters), generate samples of size $n=5,10,50,100,500$. We shall call these sets of data set 1, set 2,..., and set 5.
• For uniform distributions (with different parameters), generate samples of size $n=5,10,50,100,500$. We shall call these sets of data set 6, set 7, ..., and set 10.

For the normal distributions, you already know the variances and the standard deviation. For the uniform distributions, for each set of data, compute the real means, real variances, and real standard deviations.

Statisticians: Take your friend's generated data, and for each set of data compute the sample mean ${\bar {X}}$, the sample variance $S^{2}$, and the sample standard deviation $S$.

Compare your estimates of means and s.d. with the actual values from your generator.

Confidence intervals, known variance $\sigma ^{2}$

For each of the data set, ask your friend for the real $\sigma$.

Recall that the random variable ${\bar {X}}$ is a normal random variable with mean $\mu$ and s.d. $\sigma /{\sqrt {n}}$ Therefore,

${\sqrt {n}}({\bar {X}}-\mu )/\sigma$

will be a unit normal random variable.