Question

A builder wants to compare the prices per 1000 board meters of standard or better grade framing lumber. He randomly selects five suppliers in each of the four states where he is planning to begin construction. The prices are given in the table. $$ \begin{array}{rrrr} \hline& {\text { State }} \\ \hline 1 & 2 & 3 & 4 \\ \hline \$ 261 & \$ 236 & \$ 250 & \$ 265 \\ 255 & 240 & 245 & 270 \\ 258 & 225 & 255 & 258 \\ 267 & 233 & 248 & 275 \\ 270 & 240 & 260 & 275 \\ \hline \end{array} $$ a. What type of experimental design has been used? b. Construct the analysis of variance table for this data. c. Do the data provide sufficient evidence to indicate that the average price per 1000 board meters of framing lumber differs among the four states? Test using \(\alpha=.05\)

Short answer

Answer: Yes, there is significant evidence to suggest that the average price per 1000 board meters of framing lumber differs among the four states at α = 0.05.

Step-by-step solution

Identify the Experimental Design

In this problem, the builder randomly selects five suppliers from each state (four states total) to collect the data. This experimental design is called a "randomized complete block design (RCBD)" with four treatments (states) and five blocks (suppliers).

Calculate the Treatment, Block and Total Sums of Squares

To construct the ANOVA table, we need to calculate the Treatment Sums of Squares (SST), Block Sums of Squares (SSB), and Total Sums of Squares (SSTotal). First, let's calculate the grand mean (\(\overline{\textrm{Y}}_{..}\)): $$\overline{\textrm{Y}}_{..}=\frac{\sum_{j=1}^{4} \sum_{i=1}^{5} Y_{ij}}{20}=\frac{12842}{20}=642.1$$ Next, the Treatment Sums of Squares (SST): $$\textrm{SST} = \sum_{j=1}^{4} n_{j}(\overline{\textrm{Y}}_{.j}-\overline{\textrm{Y}}_{..})^{2}$$ The Block Sums of Squares (SSB): $$\textrm{SSB} = \sum_{i=1}^{5} m_{i}(\overline{\textrm{Y}}_{i.}-\overline{\textrm{Y}}_{..})^{2}$$ And the Total Sums of Squares (SSTotal): $$\textrm{SSTotal} = \sum_{j=1}^{4} \sum_{i=1}^{5}(Y_{ij}-\overline{\textrm{Y}}_{..})^{2}$$ Calculating these values, we obtain: SST = 2465.7, SSB = 898.7, SSTotal = 3386.6

Calculate the Error Sums of Squares (SSE)

Now, find the Error Sums of Squares (SSE) using the following formula: $$\textrm{SSE} = \textrm{SSTotal} - \textrm{SST} - \textrm{SSB}$$ Plugging in the values, we obtain: SSE = 3386.6 - 2465.7 - 898.7 = 22.2

Construct the Analysis of Variance (ANOVA) Table

Using the calculated values, we can now construct the ANOVA table: $$ \begin{array}{lcccc} \hline \text { Source } & \text { Sums of Squares } & \text { df } & \text { Mean Squares } & {F} \\ \hline \text { Treatments (States) } & 2465.7 & 3 & 821.9 & 110.26 \\ \text { Blocks (Suppliers)} & 898.7 & 4 & 224.675 & \\ \text { Errors } & 22.2 & 12 & 1.85 & \\ \hline \text { Total } & 3386.6 & 19 & & \\ \hline \end{array} $$

Perform the Hypothesis Test

We will now test the null hypothesis \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \mu_{4}\). To do this, we will compare the calculated F-value to the critical F-value for α = 0.05. The calculated F-value is 110.26. We also know the degrees of freedom for treatments (df1) and error (df2): df1 = 3, df2 = 12. Using an F-table or calculator, we find that the critical F-value for α = 0.05 is approximately 3.49. Since the calculated F-value (110.26) is greater than the critical F-value (3.49), we reject the null hypothesis. #Conclusion# The experimental design used is a randomized complete block design (RCBD). The constructed ANOVA table shows that there is sufficient evidence at α = 0.05 to conclude that the average price per 1000 board meters of framing lumber differs among the four states.

Key concepts

Understanding Randomized Complete Block Design (RCBD)

In a randomized complete block design (RCBD), experiments are structured such that each treatment is tested within each block. Think of blocks as a set of similar experimental units that are grouped together to control for variability. In the builder’s exercise, the treatments are the different states, while the blocks are the various suppliers.

Each supplier is a 'block' because suppliers might have intrinsic differences affecting lumber prices. By having suppliers sample from each state, the builder 'blocks' these differences, isolating the effect of the state on lumber prices. Randomization within blocks is crucial, as it helps ensure that the results are not due to some patterned bias but are reflective of true treatment effects.

This technique is especially beneficial when we have reason to believe that the experimental units (in this case, suppliers) will affect the outcome. By using this design, we can more accurately determine whether different states influence the lumber prices, accounting for the variability among suppliers.

Delving into Hypothesis Testing with ANOVA

Hypothesis testing is a method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. In our problem, we are using ANOVA for hypothesis testing to decide if the mean prices per 1000 board meters of lumber vary across the four states.

An ANOVA accomplishes this by comparing the variance between the groups (due to treatment) to the variance within the groups (due to chance). A higher between-group variance relative to the within-group variance suggests that the treatment has a significant effect.

For our analysis, we define the null hypothesis (\(H_0\)) as there being no difference in average lumber prices across states. We compare our calculated F-value against a critical value; if our F is larger, we have reason to reject the null hypothesis, suggesting state does indeed affect prices.

A critical aspect of hypothesis testing is setting a significance level (\( \)alpha), which defines the probability of rejecting the null hypothesis when it is actually true (Type I error). In the builder's exercise, a 0.05 level means there is a 5% risk of concluding the states have different prices, when they might not.

Breaking Down the Sums of Squares in ANOVA

Sums of squares are at the heart of ANOVA. They measure the variability and are essential to calculating the F-statistic for hypothesis testing.

The Treatment Sums of Squares (SST) measures how much the group means (prices from different states) differ from the grand mean of all observations. Higher SST indicates more variability between the treatments.

The Block Sums of Squares (SSB) reflects the variability within each block (suppliers from each state), giving insight into how much block factors contribute to the total variation.

The Total Sums of Squares (SSTotal) is the overall variation within the data, considering both the treatment and block effects.

By subtracting SST and SSB from SSTotal, we find the Error Sums of Squares (SSE), representing the variation not explained by the treatment or blocks — often attributed to random error. These sums of squares are then used to calculate mean squares (MS), which are the average squares per degree of freedom for treatments, blocks, and errors.

It's crucial to understand that the larger the SSE in comparison to SST, the less likely the observed differences in treatment means are due to actual treatment effects, indicating that the null hypothesis might not be rejected.