STA 6166 UNIT 3 Section 1 Answers
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# General Questions

1. In this section we discuss comparing means from a number of populations. Can you think of a situation where one might wish to compare variances among populations? compare upper quartiles among populations?

Example: (Variability)You are responsible for a machine that produces plate glass. For vision purposes, changes in the thickness of the glass from batch to batch is not that critical, but variability of the thickness of the glass within a batch, especially in a quarter square meter area might be very important.

Example: (Variability) You are planning to grow flowers (say lilies) for the general public. While the average height of the flower above its pot is not that important, wide variability in flower heights from pot to pot may be seen as a detrimental visual characteristic. The buyer, when viewing the flowers in the greenhouse may be quite interested in a consistent height plant.

Example: (Upper Quartile) Any product that has to fit into a specified space would be a good candidate for an upper quartile comparison. Two machines make bolts that attach the cylinder head to the main block of an automotive engine. The holes are predrilled and tapped so the bolts must fit. The bolt can be a little small for the hole and it will still work, but a bolt that is too large for the hole simply will not work and will be considered a lost product. We might compare the two machines on their ability to produce fewer numbers of "too large" bolts.

List the three assumptions under which analysis of variance procedures are developed (see page 385).

1. Additivity of Effects
2. Equality of group variances
3. Normality of residuals
4. Independence of residuals

The sample variance of the means of the multiple populations is given as. Is this the within means variance SW2 or the between means variance, SB2?

MSB = SB2

The F-test for equality of means from multiple populations if formed from the ratio of two variance estimates. Which of the test statistics below is the correct ratio?

Correct

Wrong

One column of the analysis of variance table for a completely randomized design consists of some sums of squares. Below are two column lists. In the first column are the names of the sums of squares and in the second column are the equations for the sums of squares. Match the equations with their correct names.

The names have been moved around to match the equations.

 SSB (between sample sums of squares) a. TSS (total sums of squares) b. SSW (within sample sums of squares) c.

Suppose you have performed an experiment using a completely randomized design to compare t=5 treatments. Each treatment has been replicated ni=4 times. Complete the table below with the appropriate degrees of freedom.
 Source Degrees of Freedom Between samples t-1 = 4 Within samples 19-4 = 15 Totals (5)(4)-1 -19

Using the degrees of freedom from the table above and Table 8 in the book, what is the critical value for the F test at a type I error rate of 5%?

F(0.05,4,15) = 3.06

The linear model typically used to analyze a one-way classification study (completely randomized design) has the form: . Describe in your own words what each term of the model describes about the data. For example, the m term describes the overall average for the data.

• yij - the j-th observation from the i-th group
• m - the overall mean (if all experimental units had the same treatment its average would be this.)
• ai - the (average) effect of the i-th treatment - how much the i-th treatment changes the average response from the overall mean, m.
• eij - the residual left over once we remove the overall mean and the effect of the i-th treatment from the (ij)-th observation - assumed to have common variance estimated by the MSE and zero mean.

Can you explain why the null hypothesis that all the ai terms are equal to zero is equivalent to the hypothesis that all treatment means are the same.

Think of it this way, the average response for the i-th group is m + ai. If all the ai are equal to zero, then the mean of each group is the same, m.

Below are listed typical responses that would not be expected to have homogeneous variances. Beside each response list the transformation typically used prior to the analysis of variance in an attempt to produce more homogeneous variances.
 Response Transformation Counts between 0 and 10 Square Root Counts above 10 Natural Logarithm Proportion Arcsine square root Percentage Arcsine square root of percentage divided by 100. Rate Inverse or sometimes Natural Logarithm
How does the hypothesis of the Kruskal-Wallis Test differ from the hypothesis tested with the usual analysis of variance procedures?

The Kruskal-Wallis test is a non-parametric test and as such states its null hypothesis about the equality of treatment group MEDIANS. The standard analysis of variance states it null hypothesis about the equality of treatment group MEANS (or AVERAGES).