Stats 126 CH 8 (8.1,8.2)

Sampling distribution

A probability distribution for all possible values of the statistic computed from a sample size n

Sampling distribution of the sample mean (x-bar)

The probability distribution of all possible values of the random variable x-bar computed from a sample of size n from a population with mean mu and std sigma.

Obtaining the sampling distribution of the sample mean 

1. Obtain a simple random sample of size n

2. Compute the sample mean

3. Assuming that we are sampling from a finite population, repeat steps 1 and 2 until all distinct simple random samples of size n have been obtained.

4. samples cannot be obtained a second time if already obtained

What role does n, the sample size, play I the std of the distribution of the sample mean?

As the sample size increases, the std of the sample mean decreases

Standard error of the mean

Is the std of the sampling distribution of x-bar, sigma x

Shape of the sampling distribution of x-bar if X is normal

If random variable X is normally distributed then the sampling distribution of the sample mean is approximately normally distributed

• that means mean stays the same and std becomes std error (decreases)

Saple mean for nonnormal population

1. The mean of the sampling distribution of the sample mean is equal to the mean of the underlying population, and the std of the sampling distribution of the sample mean is sigma/square root n, regardless of the size of the sample

2. The shape of the distribution of the sample mean becomes approximately normal as the sample size n increases, regardlessof the shape of the underlying population

Central limit theorem

Regardless of the shape of the underlying population, the sampling distribution of x-bar becomes approximately normal as the sample size, n, increases 

• distribution of sample mean is approximately normal provided that the sample size is greater than or equal to 30

sample proportion

denoted as p-hat and is given by p-hat=x/n

• is based on a population in which each individual either does or does not have a certain characteristic 

sampling distribution of p-hat

for a simple random sample of size n with a population proportion p:

1. the shape of the sampling distribution of p-hat is approximately normal provided np(1-p) is greater than or equal to 10

2. The mean of the sampling distribution of p-hat is mu(p)= p

3. the standard deviation of the sampling distribution of p-hat is equal to square root of p(1-p)/n

when a question asks “describe the sampling distribution” what do you find?

need to find mean and standard deviation