第8章信頼区間

“America’s Best Small Companies.” Forbes, 2013. Available online at http://www.forbes.com/best-small-companies/list/ (accessed July 2, 2013).

Data from Microsoft Bookshelf.

Data from http://www.businessweek.com/.

Data from http://www.forbes.com/.

“Disclosure Data Catalog: Leadership PAC and Sponsors Report, 2012.” Federal Election Commission. Available online at http://www.fec.gov/data/index.jsp (accessed July 2,2013).

“Human Toxome Project: Mapping the Pollution in People.” Environmental Working Group. Available online at http://www.ewg.org/sites/humantoxome/participants/participant-group.php?group=in+utero%2Fnewborn (accessed July 2, 2013).

“Metadata Description of Leadership PAC List.” Federal Election Commission. Available online at http://www.fec.gov/finance/disclosure/metadata/metadataLeadershipPacList.shtml (accessed July 2, 2013).

In many cases, the researcher does not know the population standard deviation, σ, of the measure being studied. In these cases, it is common to use the sample standard deviation, s, as an estimate of σ. The normal distribution creates accurate confidence intervals when σ is known, but it is not as accurate when s is used as an estimate. In this case, the Student’s t-distribution is much better. Define a t-score using the following formula:

The t-score follows the Student’s t-distribution with n – 1 degrees of freedom. The confidence interval under this distribution is calculated with EBM = $\left(t_{\frac{\alpha }{2}}\right)\frac{s}{\sqrt{n}}$ where $t_{\frac{\alpha }{2}}$ is the t-score with area to the right equal to $\frac{\alpha }{2}$, s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find $t_{\frac{\alpha }{2}}$ for a given α.

s = the standard deviation of sample values.

$t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}$ is the formula for the t-score which measures how far away a measure is from the population mean in the Student’s t-distribution

df = n - 1; the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

T~t_df the random variable, T, has a Student’s t-distribution with df degrees of freedom

$EBM=t_{\frac{\alpha }{2}}\frac{s}{\sqrt{n}}$ = the error bound for the population mean when the population standard deviation is unknown

$t_{\frac{\alpha }{2}}$ is the t-score in the Student’s t-distribution with area to the right equal to $\frac{\alpha }{2}$

The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by (lower bound, upper bound)
= (point estimate – EBM, point estimate + EBM)
= $\left(\overline{x}–\frac{ts}{\sqrt{n}},\overline{x}\text{+}\frac{ts}{\sqrt{n}}\right)$