1. What do confidence intervals represent? What is the most controllable method of increasing the precision of or narrowing the confidence interval? What percentage of times will the mean, or population proportion, not be found within the confidence interval? 2. As a sample size approaches infinity, how does the t distribution compare to the normal z distribution? When you draw a sample from a normal distribution, what can you conclude about the sample distribution?
(1) (a) A Confidence Interval is an interval estimate of a population parameter (such as a population mean). Instead of estimating the parameter by a single value (Point Estimate), an interval likely to include the parameter is stated. Such an interval is called a Confidence Interval. Thus, confidence intervals are used to indicate the reliability of an estimate. How likely the interval will contain the parameter is determined by the level of confidence. Increasing the level of confidence will widen the confidence interval.(b) The most controllable method of increasing the precision of a confidence interval is by increasing the sample size.
(c) Let the population mean = μ and the population standard deviation =
Assume that the confidence interval extends to Z standard deviations on both sides of the mean. Then we have
The width of the confidence interval = 2(Z * )
Percentage area inside the confidence interval = 2(Z * )%
Percentage area outside the confidence interval = [100 - 2(Z * )]%
Therefore, 100 - 2(Z * ) percentage of times the mean will not be found within the confidence interval.
[For example, if Z = 1.96, then we have the 95% confidence interval; Therefore, 100 - 95 = 5% of the times, the mean will
not be found within this confidence interval.]
(2) (a) A careful look into the t distribution probability tables, and we observe that as the number of degrees of freedom become greater than about 30 the values of the t table are very close to those of the standard normal distribution table (This is the basis for the rule of thumb of having 30 or more samples for normality -- The t- statistic approximates the z- statistic as n > 30 and approaches infinity). The t- distribution takes into account the fact that we do not know the population variance. As the number of the degrees of freedom increases then we have a better estimate of the population variance and thus the student t approaches the standard normal.
The two curves appear to the identical but there are differences. For small values of n, the curve of the t- distribution is platykurtic. The peak is narrower and the tails are fatter as compared to the normal distribution curve. This means at lower degrees of freedom, the critical t- value is higher than the critical z- value. This means the t- test is tougher and the sample evidence has to be more extreme for the null hypothesis to be rejected.
(b) When a sample is drawn from a normally distributed population, the sampling distribution is also normal.