Question Summary: -In the sample solution presented below, the expert has demonstrated our skills in answering a typical statistical question to students. The expert has explained the relationship between Sample Size and the Length of Confidence Interval for any statistical result. The expert has also demonstrated our command in explaining the concept of "Statistical Power". Solution. It is important to recognize that it is our sample size that influences the margin of error (i.e. the confidence interval). The true size of the population does not affect it. Confidence intervals from large sample sizes tend to be quite narrow in width, resulting in more precise estimates, whereas confidence intervals from small sample sizes tend to be wide, producing less precise results. We know the length of confidence interval, where n = sample size. So as the sample size increases the length of confidence interval decreases.
Now as sample size increases the length of CI decreases and converges to zero as n goes to infinity. So, the estimation becomes closer to the population parameter. Increasing sample size always improves the result but after reaching a certain sample size it is not really worth increasing our sample size any further.
Statistical power is defined as the probability of rejecting the null hypothesis while the alternative hypothesis is true. Factors that affect statistical power include the sample size, the specification of the parameter(s) in the null and alternative hypothesis, i.e. how far they are from each other, the precision or uncertainty the researcher allows for the study (generally the confidence or significance level) and the distribution of the parameter to be estimated. Statistical power is positively correlated with the sample size, which means that given the level of the other factors, a larger sample size gives greater power.
The power decreases if the length of the confidence interval is larger and increases if the length of the CI is smaller.