Statistical Inference: Estimation for Single Populations
The overall learning objective of Chapter 8 is to help you understand estimating
parameters of single populations, thereby enabling you to:
1. Know the difference between point and interval estimation.
2. Estimate a population mean from a sample mean when ( is known.
3. Estimate a population mean from a sample mean when ( is unknown.
4. Estimate a population proportion from a sample proportion.
5. Estimate the population variance from a sample variance.
6. Estimate the minimum sample size necessary to ...view middle of the document...
In an effort to understand the impact of variables on confidence intervals, it may be useful to ask the students what would happen to a confidence interval if the sample size is varied or the confidence is increased or decreased. Such consideration helps the student see in a different light the items that make up a confidence interval. The student can see that increasing the sample size, reduces the width of the confidence interval all other things being constant or that it increases confidence if other things are held constant. Business students probably understand that increasing sample size costs more and thus there are trade-offs in the research set-up.
In addition, it is probably worthwhile to have some discussion with students regarding the meaning of confidence, say 95%. The idea is presented in the chapter that if 100 samples are randomly taken from a population and 95% confidence intervals are computed on each sample, that 95%(100) or 95 intervals should contain the parameter of estimation and approximately 5 will not. In most cases, only one confidence interval is computed, not 100, so the 95% confidence puts the odds in the researcher's favor. It should be pointed out, however, that the confidence interval computed may not contain the parameter of interest.
This chapter introduces the student to the t distribution to estimate
population means from small samples when ( is unknown. Emphasize that this
applies only when the population is normally distributed. The student will
observe that the t formula is essentially the same as the z formula and that it is the
table that is different. When the population is normally distributed and ( is known, the z formula can be used even for small samples. In addition, note that some business researchers always prefer to use the t distribution when ( is unknown.
A formula is given in chapter 8 for estimating the population variance.
Here the student is introduced to the chi-square distribution. An assumption
underlying the use of this technique is that the population is normally distributed.
The use of the chi-square statistic to estimate the population variance is extremely
sensitive to violations of this assumption. For this reason, exercise extreme
caution is using this technique. Some statisticians omit this technique from
Lastly, this chapter contains a section on the estimation of sample size.
One of the more common questions asked of statisticians is: "How large of a
sample size should I take?" In this section, it should be emphasized that sample
size estimation gives the researcher a "ball park" figure as to how many to sample.
The “error of estimation “ is a measure of the sampling error. It is also equal to
the + error of the interval shown earlier in the chapter.
8.1 Estimating the Population Mean Using the z Statistic.
Finite Correction Factor