## Sampling Distribution

**Overview:**

A sampling distribution is the distribution from all proportions from all possible samples. The sampling distribution is important because of its connection with the Central Limit Theorem.

**Variability**

SD(p̂): Standard deviation for a proportion. Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of

*p̂*is modeled by a Normal model.

*p̂*: proportion of successes

*n*: sample size

SD(

*ȳ*): Standard deviation of a mean. When a random sample is drawn from any population with mean µ and standard deviation σ, its sample mean,*ȳ*, has a sampling distribution with the same*mean*µ but whose*standard deviation*is .σ : standard deviation

*n*: sample sizeCentral Limit Theorem This theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough.

The shape of any normal curve is determined by the mean and standard deviation therefore if we know both of these we can find the mean and standard deviation of the sampling distribution.[1]

Assumptions and Conditions

As the sample size gets bigger the sampling distribution model will become more normal.

The shape of any normal curve is determined by the mean and standard deviation therefore if we know both of these we can find the mean and standard deviation of the sampling distribution.[1]

Assumptions and Conditions

- Randomization Condition: The data being collected must be random.
- 10% Condition: The sample size,
*n*, must be no larger than 10% of the population. **Sampling Distribution Model for a Proportion**: Success/Failure Condition: The sample size has to be big enough so that we expect at least 10 successes and at least 10 failures.*np*> 10*nq*> 10

Large Enough Sample Condition: If the population is unimodel and symmetric even if it’s a small sample size it’s ok, but if is strongly skewed we have to have a larger sample.__Sampling Distribution Model of a Mean:__

**Final Reminder:**As the sample size gets bigger the sampling distribution model will become more normal.

**Example:**a) In the question the mean is given which is 7% and to find the standard deviation we would use the standard deviation for a proportion equation.

This should equal 0.018 = 1.8%.

b) There was a random sample of all possible clients, they represent less than 10% of all possible clients. And there were at least 10 success and failures:

c) Since our sample was collected randomly and was large, we can use the Normal Model. The mean is 0.07 and the standard deviation is 0.018. So using the normal model we find normcdf(.1,9E99,0.07,0.018) = 0.04779 or 4.78% chance.

b) There was a random sample of all possible clients, they represent less than 10% of all possible clients. And there were at least 10 success and failures:

*np*= (200)(.07) = 14*nq*= (200)(.93) = 186.c) Since our sample was collected randomly and was large, we can use the Normal Model. The mean is 0.07 and the standard deviation is 0.018. So using the normal model we find normcdf(.1,9E99,0.07,0.018) = 0.04779 or 4.78% chance.

Video from https://www.youtube.com/watch?v=0ZstEh_8bYc