Question

Four different paints are advertised as having the same dryingtime. To check the manufacturer's claims, five samples were testedfor each of the paints. The time in minutes until the paint was dryenough for a second coat to be applied was recorded. The followingdata were obtained.

Excel or Minitab users: The data set is available in file namedPaint. All data sets can be found in your eBook or on your StudentCD.

Atthe = .05 level of significance, test to see whether themean drying time is the same for each type of paint.

1. Compute the values identified below (to 2 decimals, ifnecessary).
   
   

   Sum of Squares,Treatment
   Sum of Squares,Error
   Mean Squares,Treatment
   Mean Squares,Error

   
   
   
2. Calculate the value of the test statistic (to 2 decimals).
   

Answer 1

Concepts and reason

The concept of One-way ANOVA and test statistic is used to solve this problem.

One-way Analysis Of Variance or One-way ANOVA tests whether there is any statistically significant difference between the means of three or more than three groups which are not related to each other or are independent.

A standardized value computed using the sample data during the hypothesis testing is called test-statistic.

Fundamentals

General form of the ANOVA table is defined as,

Sources of variation	Sum of squares	Degree of freedom	Mean square	F-statistic
Between treatments
Error
Total

Where SST is sum of squares due to treatment, SSE is sum of squares due to error, TSS is total sum of squares, MST is mean squares due to treatment, MSE is mean squares due to error, F is the test statistic, k denotes the number of groups or levels of treatments and n is the total number of observations.

Treatments are the factors affecting the dependent variable and the test-statistic here is nothing but the F-statistic which is obtained by dividing the mean squares due to treatment by mean squares due to error that is.

(1)

Level of significance is. To test whether there is any significant difference between different paint categories that is paint1, paint2, paint3 and paint4, the null hypothesis is defined as follows,

It says that the mean drying time is the same for each type of paint.

It says that the mean drying time is different for each type of paint.

Test for analysis of variance is done using MINITAB. Steps to perform ANOVA in MINITAB are as follows,

a. Click on StatANOVAOne-Way…, a new dialogue box will appear as shown in the next step.

Figure1

b. Select the variable time in minutes which will present there in the left box and take it into the box in front of response and do the same with the variable paint category and take it into the box in front of factor. The variables can be transferred by double clicking on them.

Figure2

c. Press “OK” button in the above dialogue box. The output obtained is shown in the screenshot below,

Figure3

The value of sum of squares due to treatment that is paint category can be obtained from ANOVA table in figure3 and is equal to.

The value of sum of squares due to error can be obtained from ANOVA table in figure3 and is equal to.

The value of mean squares due to treatment that is paint category can be obtained from ANOVA table in figure3 and is equal to.

The value of mean squares due to error can be obtained from ANOVA table in figure3 and is equal to.

Here, the treatments are paint categories. From the above screenshot it can be seen that of is greater than which depicts that may be accepted. Hence, there is no significant difference in the mean drying time of the different paint categories that is the mean drying time of all the paint categories is same.

(2)

The test statistic here is nothing but the F-statistic. The value of the F-statistic can be obtained from ANOVA table in figure3 and is equal to.

It can also be calculated using the formula. Values of MST and MSE can be obtained from part 1 and their values are. Hence,

Ans: Part 1

Sum of squares, Treatment
Sum of squares, Error
Mean squares, Treatment
Mean squares, Error

Part 2

The value of F-statistic is.