PLS205, Winter 2014Name: ______

2.What is the expected value for the Treatment1-Block1 cell? What is its residual?

Treatment / Block
Block / 1 / 2 / Means
1 / 22 / 30 / 26
2 / 26 / 38 / 32
Trtmt Means / 24 / 34 / 29

In an RCBD, the underlying linear model which explains the observed value of each and every experimental unit is

The expected (or predicted) value for every experimental unit is

(i.e. the linear model without error). And the residual for every experimental unit is

The effect of Treatment 1 is

The effect of Block 1 is

So the expected value for the Treatment1-Block1 cell is:

Expected 29-5-3 = 21

And the residual is:

ε11 = Observed Y11 – Expected Y11 = 22-21 = 1

9.In no more than two sentences, explain why the MSE may actually increase when blocks are introduced.

If there are absolutely no differences among blocks (no significant block effect), then the SSE error will be the same after the introduction of blocks (SSERCBD = SSECRD). However, because the dferror RCBD dferror CRD, the MSE will be greater in the RCBD.

11.What determines the Type I error in an experiment?

The Type I error rate in an experiment is α, which is set by the researcher.

25.For a given sample size, fixed variance, and fixed Type I error rate, briefly explain how changes in the true difference between population means will affect the probability of a Type II error. Use a diagram to help support your argument.

As means move apart the critical value stays in the same place so the area under the curve when the alternative hypothesis is true gets smaller. In general it is easier to detect a large difference than a small one. If everything else is equal we have more power (1 – β), and less Type II error (β) as the differences we are trying to detect are larger.

In the above picture, β2 < β, and (1-β2) > (1-β2), or in other words, with the new alternative mean of 76.9, the Type II error is smaller, and the power is greater.

29.A normally-distributed population has one true parametric mean and one true parametric variance. How many true parametric standard errors does it have? Refer to the definition of standard error to justify your answer.

Standard error is a function of sample size (n), as can be seen from its definition:

Since the sample size can range from 1 to N (the population size), there are N true parametric standard errors for any normally-distributed population, one for each possible sample size.