On R studio

1.Modify For the random number generator below

(a) Compute Z_i and U_i for enough values of i to cover an entire cycle

(b) Plot a scatter diagram of the Z_i values 1 apart

(c) Compute the mean value of U_i across 1 cycle

Z_i = (11Z_i-1)(mod 31),U_i =Z_i /31 with Z₀ = 7

2.Using the inverse transform method develop an algorithm for the random variable with distribution function F(x) below. Use the first three U_i valuesfrom part (a) ofproblem 1to create 3 values from therandom variable below.

F(x) = (x²)/90 < x < 3

3.Consider the following data pertaining to the amount of battery charge in a smart watch versus the number of file transfers to/from the watch.

Number of file transfersCharge remaining (hours)

28.2

46.20

65.68

84.72

104.04

123.72

153.04

(a)Determine the linear regression model for the charge as a function of the number of file transfers

(b)Calculate the coefficient of determination r²

(c)Plot the regression line and the data set and the 90% confidence interval on the regression

(d)Plotthe residuals and discuss.