Module 10: Inference for Means
Estimating a Population Mean (3 of 3)
Estimating a Population Mean (3 of 3)
Learning outcomes
- Construct a confidence interval to estimate a population mean when conditions are met. Interpret the confidence interval in context.
- Adjust the margin of error by making changes to the confidence level or sample size.
Structure of a Confidence Interval
Let’s take a closer look at the parts of the confidence interval. Remember that this is a confidence interval for a population mean. We use this formula when the population standard deviation is unknown.
Let’s remind ourselves how the confidence interval formula relates to the graph of the confidence interval on a number line.
The confidence interval shown below is a 95% confidence interval for a sample of size n = 25 (so df = 24), with sample mean [latex]\bar{x}[/latex]= 9 and sample standard deviation of s = 3. The critical T-value for a 95% confidence interval with a df = 24 is 2.064.
Standard error is [latex]3/\sqrt{25}=0.6[/latex]
Martin of error (ME) is [latex]2.064(3)/\sqrt{25}≈ 1.24[/latex]
The confidence interval is 9 ± 1.24. We are 95% confident that µ lies between 7.76 and 10.24.
Note:
- The sample mean (9 in this example) is at the center of the interval.
- The margin of error (labeled ME and equal to 1.24 in this example) is the distance that the interval extends to the left and right of the sample mean.
- The interval width is the length of the entire interval on the number line. The interval width is always twice the margin of error.
Let’s quickly review how the precision of a confidence interval relates to the margin of error:
- An interval gives a more precise estimate when the interval is narrower. In other words, the margin of error is smaller.
- An interval gives a less precise estimate when the interval is wider. In other words, the margin of error is larger.
We know that a higher confidence level gives a larger margin of error, so confidence level is also related to precision.
- Increasing the confidence in our estimate makes the confidence interval wider and therefore less precise.
- Decreasing the confidence in our estimate makes the confidence interval narrower, and therefore more precise.
Confidence interval estimates are useful when they have the right balance of confidence and precision. Typical confidence levels used in practice are 90%, 95%, and 99%. When we need to be really sure about our estimates, such as in life-and-death situations, we choose a 99% confidence level. So if nothing else changes, we settle for less precise estimates when we need a high level of confidence.
In our discussion about the structure of confidence intervals, we said choosing a higher level of confidence means that we sacrifice some precision. This is true only if nothing else changes. But there is one way to keep a high level of confidence without sacrificing precision: Increase the sample size. We investigate the impact of sample size on the confidence interval