5.4 The Behavior of Confidence Intervals
Once we know the basics of how to calculate a confidence interval, we also need to know how they behave. In other words, how does tweaking certain parts of the equation effect the interval? Keep in mind that one of the criteria that makes something a “good” statistical estimate is precision. A smaller, or more narrow, interval gives us a more precise and therefore useful estimate.
Changing the Confidence Level or Sample Size
Example
Recall the previous example:
Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of 36 scores is taken and gives a sample mean score of 68. Find a confidence interval estimate for the population mean exam score (i.e., the mean score on all exams).
Find a 90% confidence interval for the true (population) mean of statistics exam scores.
The 90% confidence interval is (67.1775, 68.8225).
Suppose we change the original problem by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.
To find the confidence interval, you need , the sample mean, and the MoE.
- = 68
- σ = 3; n = 36; the confidence level is 95% (CL = 0.95)
- MoE = () ()
Since CL = 0.95, then α = 1 – CL = 1 – 0.95 = 0.05.
= 0.025
= z0.025
The area to the right of z0.025 is 0.025, and the area to the left of z0.025 is 1 – 0.025 = 0.975.
= z0.025 = 1.96
MoE = (1.96)() = 0.98
– MoE = 68 – 0.98 = 67.02
+ MoE = 68 + 0.98 = 68.98
Notice that the MoE is larger for a 95% confidence level in the original problem, creating a less precise interval.
Interpretation: We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.
Alternative Interpretation
Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Let’s compare the results:
The 90% confidence interval is (67.18, 68.82), and the 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the figure below, you’ll see that the area 0.99 is larger than the area 0.90, so it makes sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, it is necessary for the confidence interval to be wider.
In conclusion, increasing the confidence level increases the margin of error, making the confidence interval wider.
Working Backwards to Find the Margin of Error or Sample Mean
When we calculate a confidence interval, we must first find the sample mean and calculate the margin of error. However, statistical studies may sometimes state only the confidence interval. If we know the confidence interval, we can work backward to find both the margin of error and the sample mean.
Finding the Margin of Error
- From the u