The confidence interval for estimating a population parameter has the form:
Statistic ± (critical value)*(standard deviation of statistic)
Where the statistic we use is the point estimator for the parameter.
We would always like high confidence and a small margin of error. High confidence says that our method almost always gives correct answers (high reliability). A small margin of error says that we have pinned down the parameter quite precisely (high validity). Our general formula for a confidence interval is:
Statistic ± (critical value)*(standard deviation of statistic)
With (critical value)*(standard deviation of statistic) = margin of error
Margin of error depends on the critical value and the standard deviation of the statistic. The critical value is tied directly to the confidence level: greater confidence requires a greater critical value. The standard deviation of the statistic depends on the sample size n: larger samples give more precise estimates, which means less variability in the statistic.
So the margin of error gets smaller when:
- The confidence level decreases. To obtain a smaller margin of error from the same data, you must be willing to accept lower confidence.
- The sample size increases. Increasing the sample size n reduces the margin of error for any fixed confidence level.
2 Important Cautions when constructing and interpreting confidence intervals:
- Our method of calculation assumes that the data come from an SRS of size n from the population of interest.
- The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment. Practical difficulties, such as undercoverage and nonresponse in a sample survey, can lead to additional errors that may be larger than this chance variation.
Conditions for estimating p
Before constructing a confidence interval for p, you should always check some important conditions:
- Random: The data come from a well-designed random sample or randomized experiment
- 10%: When sampling without replacement, check that n ≤ 1/10 N
- Large Counts: Both np̂ and n(1-p̂) are at least 10
When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic.