In class, we have defined the hypothesis for univariate linear regression as he(x) = 00 + 0₁x and the mean-square cost function as 1 J(00.01) - Σ (h₂(x²) — y()) ² 2m i=1 We define the following terms: m Ex0, x, y= 1Ỷy, xy= ¹ [x®y®¸ x²= ¹ [(x0)² m m m i=1 i=1 i=1 a. Show that the optimal values of the learning parameters 00 and 01 are given by: ху-ху 00 = y-x x²-x² 0₁ = ху-ху x²-x² b. What is the advantage of expressing the learning parameters in this format? =

Sum of Squared Errors: Remember from your statistics courses that if two random variables X and Y are related by a relation YaX+b and you had a set of observations {(1, 1), (2, 2)..... (z.)). then for every estimated values of a and b, the Sum of Squared Errors (SSE) was defined as in the following formula. SSE(-ar-b)². 1-1 The less SSE, the better a and b are estimated. Consider the following fixed list. In this list each sub-list of length two is standing for one pair (z.). Write a function that recieves two numbers a and b and returns the associated SSE. L- [[1, 2], [1.1, 2], [2, 7.1), (2.5, 7.21, (3, 11]]

2. Use the rbinom() function to generate a random sample of size N = 50 from the bino- mial distribution Binomial(n, p), with n 6 and p = 0.3. Note that this distribution has mean u = np and standard deviation o = Vnp(1 – p). Record the obtained sample as a vector v. Repeat the tasks of Problem 1 for the sample v.

In R, write a function that produces plots of statistical power versus sample size for simple linear regression. The function should be of the form LinRegPower(N,B,A,sd,nrep), where N is a vector/list of sample sizes, B is the true slope, A is the true intercept, sd is the true standard deviation of the residuals, and nrep is the number of simulation replicates. The function should conduct simulations and then produce a plot of statistical power versus the sample sizes in N for the hypothesis test of whether the slope is different than zero. B and A can be vectors/lists of equal length. In this case, the plot should have separate lines for each pair of A and B values (A[1] with B[1], A[2] with B[2], etc). The function should produce an informative error message if A and B are not the same length. It should also give an informative error message if N only has a single value. Demonstrate your function with some sample plots. Find some cases where power varies from close to zero to near…