Question: You have been working on a research project examining effects of fertilizer use. One of the measurements you recorded while in the field was the number of earthworms in 1 meter x 1 meter plots of soil, dug to 10 cm in depth. Two independent random samples were taken: 4 plots where fertilizer was not used, and 5 plots where fertilizer was used. (Assume the populations are normal.)

Worm counts

Sample 1 (no fertilizer) 12 3 8 5

Sample 2 (with fertilizer) 14 7 7 9 6

(a) Calculate s², the pooled estimate of \({{\sigma }^{2}}\)

(b) Find the 90% confidence interval for the difference between the two population means.

(c) Test H₀: \({{\mu }_{1}}-{{\mu }_{2}}=0\) against Ha: \({{\mu }_{1}}-{{\mu }_{2}}<0\), for \(\alpha =0.05\) (critical value approach). State your conclusions.

(d) In part c) what is the relationship we are assuming exists between fertilizer and earthworms? If you made no assumption about the relationship, but were just looking for any difference, how would your conclusion change? [NOTE: You do not have to conduct a hypothesis test in part d. Explain your answers to the two questions in words, using some numbers to support your reasoning. We will be looking for clear and concise answers.