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Problem 3. A tech startup tests 4 keyboard layouts (A, B, C, D) to see which yields faster typing speed (words per minute) for novice users. Thirty‑two (32) volunteers are randomly assigned in a completely randomized design with 8 users per layout, and each user completes one timed trial. The company analyzes the between‑layout variation with a one‑way ANOVA. The total sum of squares and treatment (factor) sum of squares are calculated as follows:  SSTotal = 1040 SSTreatment = 480  

If this experiment is analyzed as a completely randomized design (CRD), what is the error degrees of freedom?  Enter only the numeric value — no units, symbols, or text.  

If a t-test was used to analyze the data, what would be the t-statistic value? 

In a completely randomized design, the sum of squares due to error (SSE) is a measure of the variation in the values of the replicates within a factor level.

If this experiment is analyzed as a randomized complete block design (RCBD), what is the error degrees of freedom?  Enter only the numeric value — no units, symbols, or text.  

The experiment was designed and analyzed as a RCBD (Randomized Complete Block Design) with the following ANOVA results: SS Factor = 60 SS Block = 20 SS Error = 40 If this same experiment were analyzed instead as a completely randomized design (CRD), what would be the total sum of squares for the ANOVA results? Enter only the numeric value — no units, symbols, or text.

For the next two questions, use the following case as context: A popular dietary supplement is being tested for efficacy. Twenty individuals were enlisted as test subjects and each individual was randomly assigned to one of two test groups — the placebo group and the treatment group. Both groups received a pill but only 10 of those were true dietary supplements (the other 10 are placebo pills).  

If the total degrees of freedom in the ANOVA is 23, how many levels does the factor of interest have?  

Suppose that the study was run over 8 days to account for day-to-day shifts (e.g., system updates, server load, room noise, proctor differences). Each day served as a block and included one test of each layout (4 tests per day) with different participants. Assignment of participants to layouts and the within-day test order were randomized. The observations (response values) and the effect of the keyboard layout remain the same.  Only the data layout (now with blocks) changed. Given this new information, fill in the following components of the ANOVA table (numeric only; no text, symbols or units): SSFactor = [ssfactor] SSTotal = [sstotal] Degrees of freedom due to the block =[dfblock] Degrees of freedom due to error = [dferror] Total degrees of freedom = [dftotal]

Increasing the number of facilities increases facility and transportation costs but reduces inventory costs and response time.