The spread of the t distribution is a bit greater than that of the standard norm Table c gives critical values for t distributions distribution The t distributions have more Each row in the table contains critical values probability in the tails and less in the for one of the t distributions; the degrees of center than does the standard normal freedom appear at the left of the row This is true because substituting the estimates s for the fixed parameter o By looking down any column, you can check introduces more variation into the statistic that the t critical values approach the normal values as the degrees of freedom increase As the degrees of freedom k increase, the t(k) density curve approaches the N(o, 1) curve ever more closely. This happens Table t ecause s estimates o more accurately distribution as the sample size increases. So using s critical in place of o causes little extra variation when the sample is large 温需
11 21 • The spread of the t distribution is a bit greater than that of the standard normal distribution. The t distributions have more probability in the tails and less in the center than does the standard normal. This is true because substituting the estimates s for the fixed parameter introduces more variation into the statistic. σ 22 • As the degrees of freedom k increase, the t(k) 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. σ σ 12 23 Table C gives critical values for t distributions. Each row in the table contains critical values for one of the t distributions; the degrees of freedom appear at the left of the row. By looking down any column, you can check that the t critical values approach the normal values as the degrees of freedom increase. 24 Table C t distribution critical values
The t confidence intervals and tests To test the hypothesis Ho A=u based on an SRS of size n, compute the To analyze samples from normal one-sample t statistic populations with unknown o, just replace the standard deviationa/n of x by its standard error in the z procedures The z procedures then become one- mple t procedures. Use P-value or critical values from the t distribution with n- 1 degrees of freedom in place of the normal values The one-sample t procedures In terms of a variable T having t(n distribution, the P-value for a test of Ho Draw an SRS of size n from a population against having unknown mean u. Alevel C confidence interval for p is H1:x>isP(T≥1) x±t H1;<AisP(T≤ where t is the upper(1-c)/2 critical value H1:≠6 for the t(n-1)distribution
13 25 The t confidence intervals and tests • To analyze samples from normal populations with unknown , just replace the standard deviation of by its standard error in the z procedures. The z procedures then become onesample t procedures. Use P-value or critical values from the t distribution with n- 1 degrees of freedom in place of the normal values. σ σ n x s n 26 The one-sample t procedures • Draw an SRS of size n from a population having unknown mean . A level C confidence interval for is μ s x t n ± where t is the upper (1-C)/2 critical value for the t(n-1) distribution. μ 14 27 • To test the hypothesis based on an SRS of size n, compute the one-sample t statistic 0 0 H : μ = μ 0 x t s n − μ = 28 • In terms of a variable T having t(n-1) distribution, the P-value for a test of against H0 1 0 H PT t : is ( ) μ > ≥ μ 1 0 H : is ( ) μ < ≤ μ PT t 1 0 H PT t : is 2 ( ) μ μ ≠ ≥