Interpretation of Confidence Intervals: Example #1

Objectvie

This example demonstrates what is meant by the level of confidence of a confidence interval. It also shows how the confidence interval width (or the associated error term) depends on the sample size, the variability, and the confidence level.

Problem Description

High blood cholesterol increases the risk of atheroscleros, the thickening of the arteries that can reduce blood flow to the heart, brain, kidneys, etc. This increases the risk of heart attack, stroke, kidney failure, etc. The cholesterol level for adult males of a specific racial group is approximately normally distributed with a mean of 4.8 mmol/L and a standard deviation of 0.6 mmol/L.

Assumptions: The population of adult males comprising the racial group is assumed to be large and the distribution of cholesterol values in this population is assumed to be approximately normal. The error term is computed as tx(s/sqrt(n)), since the assumptions justify using the t-distribution.

Confidence Interval Applet

The Confidene Interval Applet below displays confidence intervals computed for each of 100 randomly selected samples of size n = 20 from a normal distribution with µ = 4.8 mmol/L and sigma = 0.6. Each of the 100 confidence intervals are constructed with a confidence level of 95%.

Confidence Level

The confidence level is the limiting proportion of confidence intervals that contain µ in repeated samplings from the same parent distribution. Thus, a 95% confidence level means that 95% of the intervals will contain µ in the long run. This idea can be confirmed by the above applet.

Click Recalculate in the applet 10 times. The resulting numbers of intervals covering 4.8 mmol/L (the true µ) are: 94, 97, 94, 94, 97, 96, 91, 97, 96, and 97. Your values will differ since each time the applet is displayed, redrawn, or recalculated, 100 new random samples are obtained. The number of intervals containing µ = 4.8 varied from a low of 91 to a high of 97, i.e, the proportion of intervals covering µ showed a lot of variability and ranged from 0.91 to 0.97. On the other hand, the overall proportion is 0.953 = 953/1000, i.e., 953 out of 1000 sample intervals covered µ, which is quite close to 0.95. As the number of samples is increased indefinitely, the proportion of intervals covering µ will converge to 0.95.

Effect of µ on the CI Width

Click on the µ button and change the increment to 0.1 in the resulting Dialog. Click on the right arrow button of the µ control to increase the true mean to 5.5 mmol/L. The average CI width does not change as we change the value of µ.

Effect of sigma on the CI Width

Click on the sigma button and change the increment to 0.1 in the resulting Dialog. Click on the right arrow button of the sigma control to increase the true standard deviation to 1 mmol/L. The average CI width increases as we increase the value of sigma.

Effect of n on the CI Width

Click on the left arrow button of the n control to decrease the sample size to 10. The average CI width increases as we decrease the value of n.

Effect of the Confidene Level on the CI Width

Click on the right arrow button of the alpha control to decrease the confidence level to 90%. The average CI width decreases as we decrease the confidence level.