The sampling distributions of x- for large and small samples from a normally distributed population x differ in their variability. For large samples where the sample size is much smaller than the population, the sampling distributions have approximately the same variability whether sampling is done with or without replacement. However, for small samples where the sample size represents a significant fraction of the population, sampling without replacement produces a sampling distribution with noticeably lower variability than sampling with replacement.

1. Explain the differences in the sampling distributions of x- for large and small samples under the
following assumptions. The variable of interest. x. is normally distributed. Nothing is known
about the distribution of the variable x.
Solution
The variability of a sampling distribution is measured by its variance or its standard deviation.
The variability of a sampling distribution depends on three factors:
If the population size is much larger than the sample size, then the sampling distribution has
roughly the same sampling error, whether we sample with or without displacement. On the other
hand, if the sample represents a significant fraction (say, 1/10) of the population size, the
sampling error will be noticeably smaller, when we sample without replacement.