RNA analysis of wheat genes. Engineers from the Department of Crop and Soil Sciences at Washington State University used regression to estimate the number of copies of a gene transcript in an aliquot of RNA extracted from a wheat plant (Electronic Journal of Biotechnology, April 15, 2004). The proportion (x₁) of RNA extracted from a cold-exposed wheat plant was varied, and the transcript copy number (y, in thousands) was measured for each of two cloned genes: Mn SuperoxideDismutose (MnSOD) and Phospholipase D (PLD). Letting x₁= RNA proportion and x₂= 1 if MnSOD, 0 if PLD), consider the second-order model for number of copies (y):

E(y) =₀+₁x₁+₂(x₁)²+₃x₂+₄x₁x₂+₅(x₁)²x₂

Consider this as the complete model. Data with a sample size of 36 was used to fit the complete model above using statistical software. This gave an R²value of 0.986.

1. Carry out a test of hypotheses for the overall utility of the model. Use the significance level= 0.05.

2. The researchers also want to determine if transcript copy number (y) is curvilinearly related to the proportion of RNA (x₁). Give a reduced model that is nested within the complete model so that the two models can be used to carry out a test to determine if y is curvilinearly related to x₁.

3. Give the null and alternative hypotheses for the test in 2.

4. The same data was used to fit the reduced model and it gave an R²value of 0.9858. Carry out the test of hypotheses in (b) at 5% significance level and answer the researcher's question in 2. Show all work.