Math Portal
Introductory Statistics
Section 6.3 - Confidence Intervals
You can find a confidence interval for a population proportion to show the statistical probability that a characteristic is likely to occur within that population. So what does this mean?
For example, if we have a 90% confidence interval this represents a level of certainty about our estimate. If we were to repeatedly make new estimates using exactly the exact same procedure, the confidence intervals would contain the population mean (μ) 90% of the time.
We can increase the expression of confidence in our estimate by widening the confidence interval. However, by doing so you are including a greater percentage of data. For example, the United States Census Bureau can generate 90% confidence interval for the number of people, of all ages, in poverty in the United States. This interval is 35,534,124 to 37,315,094. A 95% confidence interval on the same data generates a wider set of data, from 35,363,606 to 37,485,612.
P(35,534,124 ≤ μ ≤ 37,315,094) = 90%
P(35,363,606 ≤ μ ≤ 37,485,612) = 95%
In other words, to increase the level of confidence you must widen your prospective data values.
An interval can also be expressed in the following form:
Confidence Interval = x̄ ± margin of error
The margin of error is simply the distance from the center (μ) to either of the boundaries. For example if we were to take a look at the interval 4 ≤ μ ≤ 11, we could find that the mean is 7.5. The margin of error then is the distance from each end to the center. So the margin of error is ±3.5. The interval could then be written as 7.5±3.5, which is the same thing as 4 ≤ μ ≤ 11.
The formula to calculate a confidence interval involves using a z-score. But for our purposes, we will simply use a Confidence Limit Calculator.
You can find a confidence interval for a population proportion to show the statistical probability that a characteristic is likely to occur within that population. So what does this mean?
For example, if we have a 90% confidence interval this represents a level of certainty about our estimate. If we were to repeatedly make new estimates using exactly the exact same procedure, the confidence intervals would contain the population mean (μ) 90% of the time.
We can increase the expression of confidence in our estimate by widening the confidence interval. However, by doing so you are including a greater percentage of data. For example, the United States Census Bureau can generate 90% confidence interval for the number of people, of all ages, in poverty in the United States. This interval is 35,534,124 to 37,315,094. A 95% confidence interval on the same data generates a wider set of data, from 35,363,606 to 37,485,612.
P(35,534,124 ≤ μ ≤ 37,315,094) = 90%
P(35,363,606 ≤ μ ≤ 37,485,612) = 95%
In other words, to increase the level of confidence you must widen your prospective data values.
An interval can also be expressed in the following form:
Confidence Interval = x̄ ± margin of error
The margin of error is simply the distance from the center (μ) to either of the boundaries. For example if we were to take a look at the interval 4 ≤ μ ≤ 11, we could find that the mean is 7.5. The margin of error then is the distance from each end to the center. So the margin of error is ±3.5. The interval could then be written as 7.5±3.5, which is the same thing as 4 ≤ μ ≤ 11.
The formula to calculate a confidence interval involves using a z-score. But for our purposes, we will simply use a Confidence Limit Calculator.