Statistical inference is when we are given information about a statistic and use it to try and estimate the value of a parameter. All of what we’ve learned about over the past couple units come together to help you create viable inferences using your data. You’ve learned how to collect data, how to graph data, how to find summary values from the data, and now you’re going to find out how to use that data to make predictions/inferences.
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate.
Remember that the ideal point estimator will have no bias and low variability.
The big idea is that the sampling distribution of x̄ or p̂ tells us how close to μ or p the sample mean x̄ or proportion p̂ is likely to be. Statistical estimation just turns that around to say how close to x̄ or p̂ the unknown population mean μ or proportion p is likely to be.
All confidence intervals we use will have a form similar to this: point estimate ± margin of error
The point estimate is the best guess for the value of the unknown parameter. The margin of error, 0.10, shows how close we believe our guess is, based on the variability of the estimate in repeated SRSs of size 100. We say that our confidence level is about 95% because the interval p̂±0.10 catches the unknown parameter in about 95% of all possible samples.
Confidence interval, margin of error, confidence level
A C% confidence interval gives an interval of plausible values for a parameter. The interval is calculate from the data and has the form: Point estimate ± margin of error
When asked to interpret the confidence interval, you can say “The difference between [the point estimate] and the [true parameter] will be less than the [margin of error] in [C%] of all samples.
The confidence level C gives the overall success rate of the method for calculating the confidence interval. When asked to interpret the confidence level, you can say, “In C% of all possible samples, the method would yield an interval that captures the true parameter value.”
Important Notes
- Margin of error is NOT the same as standard error
- Standard error describes how far the sample statistic typically is from the population parameter.
- The margin of error is a multiple of the standard error, calculated so that the statistic will be within margin of error of the parameter of C% of samples
- Margin of error in a confidence interval accounts for sampling variability due only due to random selection; it does not compensate for any bias in the data collection process. If the sample wasn’t random or the questions were worded poorly, you shouldn’t have much confidence at all!
- The word “plausible” in the confidence interval box is important. “Plausible” means that if we were to do a census and actually know the value of the population parameter, we wouldn’t be surprised if it was any one of the values in the interval. We would be surprised, however, if the true value of the parameter was not one of the values in the interval, but it is still a possibility.
Interpreting Confidence Intervals
To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from ___ to ___ captures the [parameter in context].”
To say that we are 95% confident is shorthand for “If we take many samples of the same size from this population, about 95% of them will result in an interval that captures the actual parameter value.”
The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times.
However, in practice, we tend to calculate only a single confidence interval for a given situation.
The confidence level does not tell us the chance that a particular confidence interval captures the population parameter. Instead, the confidence interval gives us a set of plausible values for the parameter.
On a given problem, you may be asked to interpret the confidence interval and/or level. Be sure you understand the difference: the confidence interval gives a set of plausible values for the parameter and the confidence level describes the long-run capture rate of the method. When interpreting an interval, make it clear that you are describing a parameter, and not a statistic. Be sure to include context!
As confidence level increases, interval size increases/widens. As confidence level decreases, interval size decreases/narrows.
The critical value is a multiplier that makes the interval wide enough to have the stated capture rate. The critical value depends on both the confidence level C and the sampling distribution of the statistic.