Business Statistic Question (help me, explain step by step)

Question

A town that recently started a single-stream recycling program provided 60-gallon recycling bins to 13 randomly selected households and 75-gallon recycling bins to 11 randomly selected households. The total volume of recycling over a 10-week period was measured for each of the households. The average total volumes were 365 and 423 gallons for the households with the 60- and 75-gallon bins, respectively. The sample standard deviations were 52.5 and 43.8 gallons, respectively. Assume that the 10-week total volumes of recycling are approximately normally distributed. Using the 2% significance level, can you conclude that the variance for the 10-week recycling volume of all households having 60-gallon containers is different from the variance of all households that have 75-gallon containers?

Answer 1

For the ratio of sample variances, we use the F statistic with 13-1 and 11-1 df = 12 and 10 df.

As α = 2% and we are testing for differences, our critical values are when P(F statistic or less)= .01 and .99

In Excel, F.INV(0.01,12,10) = .233 and F.INV(0.99,12,10) = 4.706

The ratio of sample variances is 52.5^2/43.8^2 = 1.437. As this is in the interval (.233, 4.706), we fail to reject the null hypothesis that the variances are the same.