Question

Suppose that a random sample of size \(n=5\) was selected from the vineyard properties for sale in Sonoma County, California, in each of three years. The following data are consistent with summary information on price per acre (in dollars, rounded to the nearest thousand) for disease-resistant grape vineyards in Sonoma County (Wines and Vines, November 1999). $$\begin{array}{llllll} 1996 & 30,000 & 34,000 & 36,000 & 38,000 & 40,000 \\ 1997 & 30,000 & 35,000 & 37,000 & 38,000 & 40,000 \\ 1998 & 40,000 & 41,000 & 43,000 & 44,000 & 50,000 \end{array}$$ a. Construct boxplots for each of the three years on a common axis, and label each by year. Comment on the similarities and differences. b. Carry out an ANOVA to determine whether there is evidence to support the claim that the mean price per acre for vineyard land in Sonoma County was not the same for the three years considered. Use a significance level of \(.05\) for your test.

Short answer

The boxplots will give a visual representation of the distribution, center and spread of the prices for each year. The outcome of the ANOVA test will give evidence whether to accept or reject the claim that the mean prices are the same for all three years.

Step-by-step solution

Construct Boxplots

Boxplots can be constructed using any statistical software or even manually. The minimum, first quartile, median, third quartile, and maximum are calculated for each year and a boxplot is sketched for the computed values. Draw your scale and create a box with three vertical lines that represent the first quartile, median and third quartile for the price values. Finally, create lines (whiskers) to the minimum and maximum value, originating from the box.

Comment on Boxplots

Study the boxplots constructed. Comment on the range, interquartile range, median, any possible outliers and similarities and differences between the plots for each year. The interpretation will be based on the distribution, center and spread of the boxplots constructed.

Calculate ANOVA

Firstly, calculate the sum of squares within (SSW) and sum of squares between (SSB). The SSW is the variation due to differences within each year, while the SSB is the variation due to differences between years. Next, the degrees of freedom between (dfb) and degrees of freedom within (dfw) are calculated. Then, The Mean of Sum Squares Between (MSB) and Mean of Sum Squares Within (MSW) are calculated. Finally, the F statistic is calculated by dividing MSB by MSW.

Hypothesis Testing

Set up the null hypothesis that all means are equal and the alternative hypothesis that at least one mean is different. The computed F statistic value is then compared with the F critical value from the F distribution table for dfb and dfw at the significance level of 0.05. If the calculated F statistic is greater than the F critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.