Question

Define sampling distribution. b. Explain why the spread (variance) of a sampling distribution for estimating a mean would be smaller than the spread (variance) of the whole population.

c. Discuss what happens to the shape AND spread of a sampling distribution as sample size (n) is increased.

d.Explain why it is essential that the sampling distribution be normally distributed (or t-distributed) in order to do the statistical inference procedures we use in Chapters 9, 10 and 11.

Answer 1

a. Sampling Distribution : A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drwan from a specific population.

Example: Height of a males of USA is the population here. Now we are taking samples of males from different different states. So here samples which we are drawing from the population can be said as the subsets of the population. Now each sample has its own sample mean and the distribution of the sample means is known as sample distribution. Now the average height computed for each sample set can be called as sampling distribution of the mean. Point to note that, Sample Distribution Sampling Distribution

b. Taking the previous example of male heights of USA, the variance or spread of the whole population will be much larger because it consists of a large population. In a nutshell it can be said that, within a population the variability of heights will be much more higher. On the other hand, within the sample means that are not variable very much. Hence the spread of population is much higher than the spread of sample means (sampling distribution)

c. When sample size (n) is increased the spread of the sampling distribution will be reduced. And the shape will be more dense around the mean of the true population.  

d. From the Central Limit Theorem, we know that the sampling distribution of the mean of any random, independent variables will be normal or nearly normal, if the sample size is large enough. When the population distribution is nearly normal distribution then the minimun sample size of 30 is well enough. But if the population distribution is highly skewed, then the sample size need to be increased to get a normally distributed sampling distribution.

Now, if the population standard deviation is known, we use normal distribution. and if the population standard deviation is unknown, we use T distribution.