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Unformatted text preview: Stats Study Guide Exam 4 Chapter 18: Inference about a Population Mean Online Notes Here are the conditions needed for realistic inference about a population mean: CONDITIONS FOR INFERENCE ABOUT A MEAN We can regard our data as a simple random sample (SRS) from the population. This condition is very important. Observations from the population have a Normal distribution with mean and standard deviation . In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small. Both and are unknown parameters. Because is unknown, STANDARD ERROR When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic. The standard error of the sample mean is s / n . And THE ONE-SAMPLE t STATISTIC AND THE t DISTRIBUTIONS has the t distribution with n 1 degrees of freedom. Degrees of freedom : there is a different t distribution for each sample size. We specify a particular t distribution by giving its degrees of freedom. (n 1 in for t distributions) Differences b/w t distribution density curve and a normal curve The density curves of the t distributions are similar in shape to the standard Normal curve. They are symmetric about 0, single-peaked, and bell-shaped. The spread of the t distributions is a bit greater than that of the standard Normal distribution . The t distributions in Figure 18.1 have more probability in the tails and less in the center than does the standard Normal. This is true because substituting the estimate s for the fixed parameter introduces more variation into the statistic . As the degrees of freedom increase, the t density curve approaches the N (0,1) curve ever more closely. This happens because s estimates more accurately as the sample size increases. So using s in place of causes little extra variation when the sample is large. o THE ONE-SAMPLE t CONFIDENCE INTERVAL Stats Study Guide Exam 4 Draw an SRS of size n from a large population having unknown mean . A level C confidence interval for is where t * is the critical value for the t ( n 1) density curve with area C between t * and t * ....
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- Fall '08