Download Small-Sample Inference: t-Statistics for Mean Tests & Confidence Intervals and more Exams Economics in PDF only on Docsity! ECON 413 Fall 2004 Small-sample Confidence Intervals and Tests of Hypothesis for the Mean The confidence intervals and test procedures we have used so far assumed that the value of the population standard deviation σ was known (or alternatively the sample size was assumed so large so that the sample standard deviation s provided a good estimate for σ). When the sample size is small and the value of σ is unknown, s will not be a good estimate of σ. Therefore, we can no longer use the estimation and testing procedures based on the normal distribution. Our small-sample inference procedures are based on the t-statistic / xt s n µ− = This statistic differs from the usual z-statistic by the appearance of s in place of σ. However, the distribution of this statistic is not N(0,1). This statistic has a distribution called t-distribution with (n-1) degrees of freedom when our sample is a simple random sample from a normal population. The degrees of freedom is a parameter of the t- distribution. The distribution changes as the degrees of freedom changes. The t- distribution has many similarities to the normal distribution. It is symmetric, bell-shaped, and its mean and median are equal. However, the t-distribution has fatter tails than the normal distribution. That is, there is more probability in the tails of the distribution. As the degrees of freedom goes to infinity, the t-distribution tends to the normal distribution. The areas under the t-distribution can be calculated by using a special table. We shall denote a t-distribution with k degrees of freedom by t(k). A level C small-sample confidence interval for µ is calculated as * sx t n ± × where t* is the value from t(n-1) density curve such that the area under the density between –t* and t* is C. This formula is valid only if the population is normally distributed. The formula will be approximately correct if the sample size is large. If we wish to test a hypothesis about the mean µ, say H0: µ = µ0, we can do this by calculating the value of the t-statistic and the P-value. The P-value is calculated 0 0 0 : , ( ) ( : , ( ) ( : , 2 ( ) ( a a a if H then P value P T t if H then P value P T t if H then P value P T t 1) 2) 3) µ µ µ µ µ µ > − = ≥ < − = ≤ ≠ − = × ≥ These P-values are exact if the population distribution is normal and otherwise they are approximately correct if n is large.
