The model yi = Bo +B1Xi + i, where &i ~ N(0,0%) and independent, is to be used to predict the annual consumption of a product, y, as a function of gender where xi = 1 if the i-th individual is female, and 0 if the individual is male. A sample of n individuals is selected, n, of which are female, and the remaining n, are male. For each of these n individuals, the annual amount spent on the product, y, is recorded. (a) Assuming that the data set is such that the n, observations for females appear first, followed by the na observations for males, specify the y vector and X matrix. (b) Determine the vector b for the least squares estimates. Describe the estimates in terms of the sample mean amounts spent by females and males. (c) Starting from the expression for the estimate of the error term variance in simple linear regression, 52 = P=1Vi - A)?/(n 2), show that s2 = [(n 1)s + (nz 1)s]/(n +n1 2), where s is the sample variance of the y observations for females, and s is the sample variance of the y observations for males. Note that sa here is simply the pooled estimate of variance for a t-test that compares the means of two normally distributed populations that are assumed to have equal variance. The model yi = Bo +B1Xi + i, where &i ~ N(0,0%) and independent, is to be used to predict the annual consumption of a product, y, as a function of gender where xi = 1 if the i-th individual is female, and 0 if the individual is male. A sample of n individuals is selected, n, of which are female, and the remaining n, are male. For each of these n individuals, the annual amount spent on the product, y, is recorded. (a) Assuming that the data set is such that the n, observations for females appear first, followed by the na observations for males, specify the y vector and X matrix. (b) Determine the vector b for the least squares estimates. Describe the estimates in terms of the sample mean amounts spent by females and males. (c) Starting from the expression for the estimate of the error term variance in simple linear regression, 52 = P=1Vi - A)?/(n 2), show that s2 = [(n 1)s + (nz 1)s]/(n +n1 2), where s is the sample variance of the y observations for females, and s is the sample variance of the y observations for males. Note that sa here is simply the pooled estimate of variance for a t-test that compares the means of two normally distributed populations that are assumed to have equal variance