Inference for the Mean of a Population

Inference for The goal is to make inference about is a point estimate of . If we sample from a normal distribution, the sampling distribution of is standard normal.   If we sample from a non-normal distribution, the sampling distribution above is approximately standard normal for large n.

Standard Error is seldom known Estimate by S When the standard deviation of a statistic is estimated from the data, the result is called the standard error of a statistic. The standard error of the sample mean is

The t Distributions Suppose that an SRS of size n is drawn from a population. Then the one-sample t statistic has the t distribution with n – 1 degrees of freedom. The assumption that we sample from a normal population is important for small n, but not for large n.

Properties of the t Distribution continuous and symmetric about 0 more variable and slightly different shape than standard normal (see Figure page 494) There are an infinite number of t distributions. They are indexed by the number of degrees of freedom. As the number of degrees of freedom increase, the t distribution approaches the standard normal.

The One-Sample t Confidence Interval Suppose that an SRS of size n is drawn from a population having unknown mean . A level C confidence interval for is Where t* is the value for t(n – 1) density curve with area C between –t* and t*. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.

Margin of Error The margin of error for the population mean when we use the data to estimate is

The One-Sample t Test Suppose that an SRS of size n is drawn from a population having unknown mean . To test the hypothesis Ho: based on an SRS of size n, compute the one-sample t statistic