Artificial Teeth: Wear. In a study by J. Zeng et al. three materials for making artificial teeth-Endura, Duradent, and Duracross-were tested for wear. Their results were published as the paper "In Vitro Wear Resistance of Three Types of Composite Resin Denture Teeth" (Journal of Prosthetic Dentistry, Vol. 94 . Issue 5. pp. 453-457 ). Using a machine that simulated grinding by two right first molars at 60 strokes per minute for a total of 50,000 strokes, the researchers measured the volume of material worn away, in cubic millimeters. Six pairs of teeth were tested for each material. The data on the WeissStats site are based on the results obtained by the researchers. At the 5 % significance level, do the data provide sufficient evidence to conclude that there is a difference in mean wear among the three materials?

• Examine whether the figures provide enough evidence to prove that the three materials have distinct mean wear rates.
• It is necessary to state the null and alternative hypotheses.
• Nullity hypothesis:

• $H_0:{\mu }_{\text{Endura}}={\mu }_{\text{Duradent}}={\mu }_{\text{Duracross}}$

• That is, the data is insufficient to determine whether the three materials have distinct average wear rates.

• Another option is:

• At least one${\mu }_i$ isn't the same as the others.

• In other words, the numbers are sufficient to show that the three materials have differing mean wear rates.

• The significance level is $\alpha$=0.05.

Calculate the test statistic's value.

Procedure for MINITAB:

Step 1: Select Stat > ANOVA > One-Way from the drop-down menu.

Step 2: In the Response box, type VOLUME in the column.

Step 3: In Factor, type MATERIAL in the MATERIAL column.

Step 4: Click the OK button.

MINITAB output:

One-way ANOVA: VOLUME versus MATERIAL

The value of F is 341.16 text, while the value of p text is 0.000 , according to the MINITAB output.

Obtain the normal probability plot for residual and residual versus fits.

MINITAB procedure:

Step 1: Choose Stat > ANOVA > One-Way.

Step 2: In Response, enter the column of VOLUME.

Step 3: In Factor, enter the column of MATERIAL.

Step 4: In Graph, Choose normal probability of residual and residual versus fits.

Step 5: Click OK.

MINITAB produces text. The plot of residual normal probability

• Because the points are closer to the straight line in the normal probability plot, the data distribution is approximately normally distributed.
• The points are hence linear. Furthermore, the figure contains no outliers.
• As a result, the normalcy assumption is unaffected.

MINITAB output: Residual versus fits ;

The largest to smallest standard deviation ratio is,

$\text{Ratio}=\frac{0.08}{0.03}$

$=2.67$

• The highest to lowest standard deviation ratio is more than two.
• This shows that the equal standard deviation assumption has been broken.
• As a result, it is reasonable to assume normal populations, but not equal population standard deviations.