Interpreting confidence intervals
Students of statistics tend to have trouble articulating just what it is a confidence interval is and does. What follows is a selection of many statements providing possible interpretations for confidence intervals.
A proper assessment of majority of these statements can be made by the average student who has paid attention in class or to the exposition developing confidence intervals in the text. It may be best to investigate some of the issues through data analysis and contemplating the logic that the textbook and instructor have used in developing and discussing these intervals.
Assume a proper random sample of ball-bearings are studied; measuring the diameters of each results in a 95% confidence interval of 15.22 mm ± 0.17 mm – from 15.05 mm to 15.39 mm.
Which – if any – are true and which are not? For all that are false, explain why they are false.
· 95% of the sampled ball-bearings have diameter between 15.05 and 15.39. (You can use any data set to assess this interpretation. Input some data, obtain the interval, and check whether the confidence level matches the number of observations lying within the bounds of the interval.)
· There’s a 95% probability that a single future randomly sampled ball-bearing is between 15.05 and 15.39.
· The probability is 95% that the sample mean of the studied ball-bearings is between 15.05 and 15.39.
· For 95% of all other possible random samples, the sample mean will fall between 15.05 and 15.39.
· If the study were repeated under identical conditions, the probability would be 95% that the mean for the repeat sample would be between 15.05 and 15.39.
· There is a 95% probability that the mean of all ball bearings is between 15.05 and 15.39.
A student gathers data from 35 friends and uses the rule discussed in class to obtain a 95% confidence for the mean GPA. Should we have 95% confidence in this interval? More than 95%? Less than 95%? Why?
An industrial company manufactures a bearing that has desired mean width 1.42 cm; the actual widths vary slightly due to production irregularities. The distribution of widths is known to be approximately normal (this information is obtained using accumulated data from many years of quality control measurements). Each week, 8 bearings are randomly selected and measured. A 99% confidence interval is used to assess whether on average the bearing widths are meeting the standard. Consider this week’s sample and the resulting confidence interval: n = 8 seems like an awfully small sample size, doesn’t it? Does this invalidate the confidence of 99%? After all, 99% of 8 is 7.92 – almost all the bearings. If not, what is the consequence of the small sample size?
A Public Interest Group obtains information on the benefit of President Bush’s proposed tax cut. A random sample of U.S. taxpayers is used to obtain a 95% confidence interval for the median benefit: ($74, $156). Interpret this interval. Also, explain why the median is used here.
Properties of confidence intervals
The rule discussed in class for obtaining a confidence interval have these properties embedded in them – however, you must be alert not only to how the rule works in a specific case, but instead how it adjusts to handle different situations.
Questions
Explain what happens to the error margin when
· The sample size for a study is increased (but everything else remains essentially the same)? Decreased?
· The confidence level for an interval is increased? Decreased?
Due to irregularities in the production line, extra large boxes of a laundry detergent vary in weight from box to box. The average weight is somewhere around 12 pounds. Adult tomcats of a particular breed also vary in weight – the average weight for this breed is somewhere around 8 pounds. Suppose we endeavored to conduct 2 independent studies; one of the detergent boxes, one of the tomcats. Assume we sample equal numbers of each, and obtain a 95% confidence interval for the mean for each variable. Which interval (if either) has smaller error margin? Why?
Properties and interpretations of confidence can, for the most part, be generalized. Although there are many methods for obtaining (computing) confidence intervals, different ways in which randomization might take place, and a variety of parameters that one might estimate, for the most part all confidence intervals are interpreted the same way and have the same properties.
Confidence Interval Interpretation and Properties