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Estimation and Confidence Intervals for AP Statistics (page 2)

By — McGraw-Hill Professional
Updated on Feb 4, 2011

General Form of a Confidence Interval

A confidence interval is composed of two parts: an estimate of a population value and a margin of error. We identify confidence intervals by how confident we are that they contain the true population value.

A level C confidence interval has the following form: (estimate) ± (margin of error). In turn, the margin of error for a confidence interval is composed of two parts: the criticalvalue of z or t (which depends on the confidence level C) and the standard error. Hence, all confidence intervals take the form: (estimate) = (margin of error) = (estimate) ± (critical value)(standard error).

A t confidence interval for μ would take the form:

t* is dependent on C, the confidence level; s is the sample standard deviation; and n is the sample size.

The confidence level is often expressed as a percent: a 95% confidence interval means that C = 0.95, or a 99% confidence interval means that C = 0.99. Although any value of C can be used as a confidence level, typical levels are 0.90, 0.95, and 0.99.

IMPORTANT: When we say that "We are 95% confident that the true population value lies in an interval," we mean that the process used to generate the interval will capture the true population value 95% of the time. We are not making any probability statement about the interval. Our "confidence" is in the process that generated the interval. We do not know whether the interval we have constructed contains the true population value or not—it either does or it doesn't. All we know for sure is that, on average, 95% of the intervals so constructed will contain the true value.

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example: Floyd told Betty that the probability was 0.95 that the 95% confidence interval he had constructed contained the mean of the population. Betty corrected him by saying that his interval either does contain the value (P = 1) or it doesn't (P = 0). This interval could be one of the 95 out of every 100 on average that does contain the population mean, or it might be one out of the 5 out of every 100 that does not. Remember that probability values apply to the expected relative frequency of future events, not events that have already occurred.

example: Find the critical value of t required to construct a 99% confidence interval for a population mean based on a sample size of 15.

solution: To use the t distribution table (Table B in the Appendix), we need to know the upper-tail probability. Because C = 0.99, and confidence intervals are two sided, the upper-tail probability is

.

Looking in the row for df = 15 – 1 = 14, and the column for 0.005, we find t* = 2.977. Note that the table is set up so that if you look at the bottom of the table and find 99%, you are in the same column.

Using the newer version of the TI-84, the solution is given by invT(0.995,14) = 2.977.

example: Find the critical value of z required to construct a 95% confidence interval for a population proportion.

solution: We are reading from Table A, the table of Standard Normal Probabilities. Remember that table entries are areas to the left of a given z-score. With C = 0.95, we want

in each tail, or 0.975 to the left if z*. Finding 0.975 in the table, we have z* = 1.96. On the TI-83/84, the solution is given by invNorm(0.975) = 1.960.

Practice problems for these concepts can be found at:

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