Exam 16: Analysis of Variance

A partial ANOVA table in a randomised block design is shown below. Source of Variation SS df MS F Treatments * 3 * * Blocks 1256 2 * * Error * * 67.67 Total 2922 11 Can we infer at the 5% significance level that the block means differ?

In one-way ANOVA, the total variation SS(Total) is partitioned into three sources of variation: the sum of squares for treatments (SST), the sum of squares for blocks (SSB) and the sum of squares for error (SSE).

In a single-factor analysis of variance, MST is the mean square for treatments and MSE is the mean square for error. The null hypothesis of equal population means is likely false if:

The strength of a weld depends to some extent on the metal alloy used in the welding process. A scientist working in the research laboratory of a major automobile manufacturer has developed three new alloys. In order to test their strengths, each alloy is used in several welds. The strengths of the welds are then measured, with the results shown below. Can the scientist conclude at the 5% significance level that differences exist among the strengths of the welds with the different alloys? $\quad$$\quad$$\quad$$\text { Strength of welds }$ Alloy 1 Alloy 2 Alloy 3 15 17 25 23 21 27 16 19 24 29 25 31 28 23 19

The following statistics were calculated based on samples drawn from four normal populations. Treatment Statistic 1 2 3 4 4 7 5 5 52 69 71 61 753 798 1248 912 Test at the 5% level of significance to determine whether differences exist among the population means.

In a completely randomised design, 15 experimental units were assigned to each of four treatments. Fill in the blanks (identified by asterisks) in the partial ANOVA table shown below. Source of Variation SS df MS F Treatments * * 240 * Error * * * Total 2512 *

A partial ANOVA table in a randomised block design is shown below. Source of Variation SS df MS F Treatments * 3 * * Blocks 1256 2 * * Error * * 67.67 Total 2922 11 Fill in the missing values (identified by asterisks) in the above ANOVA table.

The following statistics were calculated based on samples drawn from three normal populations: Treatment Statistic 1 2 3 n 10 10 10 95 86 92 s 10 12 15 Set up the ANOVA table and test at the 5% level of significance to determine whether differences exist among the population means.

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics are calculated: $n _ { 1 } =$ 6, $\bar { x } _ { 1 } =$ 50, $s _ { 1 } =$ 5.2. $n _ { 2 } =$ 8, $\bar { x } _ { 2 } =$ 55, $S _ { 2 } =$ 4.9 . $n _ { 3 } =$ 6, $\bar { x } _ { 3 } =$ 51, $s _ { 3 } =$ 5.4. The grand mean equals:

In single-factor analysis of variance, if large differences exist among the sample means, it is then reasonable to:

The sum of squares for treatments stands for the between-treatments variation.

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: $n _ { 1 } =$ 7, $\bar { x } _ { 1 } =$ 65, $s _ { 1 } =$ 4.2. $n _ { 2 } =$ 8, $\bar { x } _ { 2 } =$ 59, $S _ { 2 } =$ 4.9. $n _ { 3 } =$ 9, $\bar { x } _ { 3 } =$ 63, $s _ { 3 } =$ 4.6. The value of the test statistics, F, equals:

Which of the following is true of the F-distribution?

The randomised block design is also called the two-way analysis of variance.

Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (4, 8) is 5.05.

A recent college graduate is in the process of deciding which one of three US graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { GMAT Scores }$ School 1 School 2 School 3 650 510 590 620 550 510 630 700 520 580 630 500 710 600 490 690 650 530 Use Tukey's method with $\alpha$ =0.05 to determine which population means differ.

A survey will be conducted to compare the grade point averages of US high-school students from four different school districts. Students are to be randomly selected from each of the four districts and their grade point averages recorded. The ANOVA model most likely to fit this situation is:

The number of degrees of freedom for the numerator or MST is 3 and that for the denominator or MSE is 18. The total number of observations in the completely randomised design must equal 22.

The F-test of the randomised block design of the analysis of variance has the same requirements as the independent-samples design; that is, the random variable must be normally distributed and the population variances must be equal.

One-way ANOVA is applied to independent samples taken from four normally distributed populations with equal variances. If the null hypothesis is rejected, then we can infer that: