MINITAB Stat Lab 3

Transcription

1 MINITAB Stat Lab 3. Statitical Inference In the previou lab we explained how to make prediction from a imple linear regreion model and alo examined the relationhip between the repone and predictor variable. Today we will take a look at the output from the ANOVA table in the eion window. We have fitted a line to the data and een whether one variable tend to increae or decreae a the other variable increae. However, we don t know whether the variable really have a linear relationhip the random variation in the ample might jut be making it look that way. To find out which i a more likely explanation, we are going to conduct a hypothei tet. Thi mean comparing the amount of variation explained by the linear model with the etimate of background or ampling variation. To do thi, we will contruct an ANOVA (Analyi of Variance) table. The hypothei being teted that the lope of the line = 0, veru an alternative that the lope of the line i not = 0. H 0 : b = 0 v H A: b 0 The tet tatitic i freedom. MS( regreion) MS( Error) and i ditributed a F with, n-2 degree of That i, IF there i no relationhip between the variable (i.e. the null hypothei i true), the tet tatitic calculated come from an F ditribution with, n-2 degree of freedom. Uing the fire damage dataet from lat week fit a imple linear regreion model. You hould ee the following output in the eion window.

2 Regreion Analyi: Damage - $ veru Ditance The regreion equation i Damage - $ = Ditance Regreion Line S = R-Sq = 92.3% R-Sq(adj) = 9.8% Analyi of Variance Source DF SS MS F P Regreion Error Total ANOVA Table Summarie the hypothei that i being teted in the ANOVA table, including Ho, Ha, the tet tatitic, the p value and tate your concluion. Take α = For each of the dataet ued in lat week lab, conduct a hypothei tet for the lope uing the ame criteria a above

3 2. Standard Error of the Slope There i another hypothei tet for the lope we can ue, which i more general than the F tet in the ANOVA above. We can calculate a tandard error of the lope uing the, which i our etimate of σ. Thi will allow u to tet hypothee about the lope uing a t-tet but we can alo u thi tandard error to get a confidence interval for the lope. Summary from Lecture Note σ The tandard error of the lope i σ ˆ β = which i etimated a = ˆβ A hypothei tet for the lope One-Tailed tet Ho: β = β 0 Ha: β < β 0 (or β > β 0 ) Two-Tailed tet Ho: β = β 0 Ha: β β 0 Tet tatitic: Rejection Region: ˆ β β t= ˆ β 0 ˆ β β = 0 One-Tailed tet Two-tailed tet t < -t α t > t α/2 (or t < t α ) Where t α and t α/2 are baed on (n-2) degree of freedom.

4 Uing the firedamage dataet, go to, Stat - Regreion - Regreion.... Select the repone 2. Select the predictor Minitab Output Regreion Analyi: Damage - $ veru Ditance The regreion equation i Damage - $ = Ditance Standard error Predictor Coef SE Coef T P Contant Ditance What i the tandard error of the lope? MINITAB by default tet the two-tailed null hypothei that the lope i zero. Report the reult of thi hypothei tet in the uual way (ue α =.0). Calculate the quare root of the F tet tatitic from the ANOVA table. What i the reult? What do you notice when you compare thi value to the value of the t tet tatitic for teting the lope i zero?

5 Repeat thi tet for each of the example in the correlation dataet tating clearly H 0, H A, α (ue α = 0.05), the tet tatitic, the p value and your concluion Confidence Interval for the Slope A confidence interval for the lope may be obtained by uing the etimated tandard error of the lope and an appropriate quantile from the t ditribution with n-2 degree of freedom. Summary from Lecture Note ˆ β ± t α / 2 S ˆ β where the etimated tandard error of ˆβ i calculated by S = ˆβ and t α/2 i baed on (n-2) degree of freedom. Aumption: Where ε i defined a( yi yˆ i ),. The mean of the probability ditribution of ε i The variance of the probability ditribution of ε i equal at all value of the predictor variable x. 3. The probability ditribution of ε i normal. 4. The value of ε aociated with any tow value of y are independent.

6 Uing the tandard error from part 2, and either the INVCDF command or the Cambridge table to get the appropriate quantile from the t ditribution to calculate the 99% confidence interval for the lope: What i the confidence interval? ( to ) REVISION SUMMARY After thi lab you hould be able to: - Calculate the correlation coefficient by hand and in Minitab - Fit a imple linear regreion line to data uing Minitab - Undertand the hypothei in the imple linear regreion ANOVA table - Tet if the lope of the model i equal to zero or not - Contruct a confidence interval for the lope