REALLY NEED HELP WITH QUESTIONS 13, 14, AND 15 PLEASE PROVIDE DETAILED EXPLANATIONS WITH STEP BY STEP DETAILS AND ANSWER ALL QUESTIONS PLEASE HELP Note The data and reference for week7 X txt and week7 Y txt is located at the bottom of the page 13 A certain golf ball brand (Brand A) claims their ball can be hit a greater distance than a competitor brand (Brand B) off of a tee To test this claim, the brands invited 16 golfers to hit both balls, and their results are listed below Assume the differences of the paired A distances and B distances are approximately normally distributed and test the null hypothesis H 0 D 0 against the null hypothesis H 1 D 0 using a paired t test with the 16 differences Let 0 05 Golfer Brand A Distance Brand B Distance 1 265 252 2 272 276 3 246 243 4 260 246 5 274 275 6 263 246 7 255 244 8 258 245 9 276 259 10 274 260 11 274 267 12 269 267 13 244 251 14 212 222 15 235 235 16 254 255 14 A company is comparing a new adhesive to an old adhesive Let X and Y denote the amount of pressure the new and old adhesives can withstand, respectively (a) Based on n m 24 observations from both adhesives, define the test statistic and critical region for testing H 0 X Y 0 against H 1 X Y 0 where 0 05 (b) For this part you will want to use the data in the files week7 X txt and week7 Y txt to calculate the value of the test statistic and state your conclusion (c) What is the approximate p value of this test (d) You'll need to construct two box plots on the same graph Do the box plots support your conclusion The next problem deals with a topic called Monte Carlo simulations In terms of machine learning, the phrase Monte Carlo generally means to run random simulations to estimate probability We'll use it below to estimate the value of an integral Monte Carlo simulations (and the Monte Carlo Tree Search Algorithm) were used by Google's company DeepMind to help train a neural network in creating the computer programs AlphaGo and AlphaZero, which have become the leading players of the games of chess, shogi, and go Their creation of AlphaGo is the first computer program to defeat a professional human Go player, the first to defeat a Go world champion, and is arguably the strongest Go player in history 15 Use Python (and NumPy) for this particular problem Consider a function f(x) which is continuous on the interval a, b Assume that m f (x) M for every x a, b for some constants m and M We've already seen how we can estimate the integral a b f ( x ) d x using Riemann sums, and other methods like the Trapezoid Rule, Simpson's Method, and Taylor series We are going to see how randomly choosing points (x,y) can estimate the integral too Consider the picture below

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REALLY NEED HELP WITH QUESTIONS 13, 14, AND 15. PLEASE PROVIDE DETAILED EXPLANATIONS WITH STEP-BY-STEP DETAILS AND ANSWER ALL QUESTIONS... PLEASE HELP!!! Note: The data

REALLY NEED HELP WITH QUESTIONS 13, 14, AND 15. PLEASE PROVIDE DETAILED EXPLANATIONS WITH STEP-BY-STEP DETAILS AND ANSWER ALL QUESTIONS... PLEASE HELP!!! Note: The data and reference for week7-X.txt and week7-Y.txt is located at the bottom of the page

13. A certain golf ball brand (Brand A) claims their ball can be hit a greater distance than a competitor brand (Brand B) off of a tee. To test this claim, the brands invited 16 golfers to hit both balls, and their results are listed below. Assume the differences of the paired A distances and B distances are approximately normally distributed and test the null hypothesis H₀ : _D = 0 against the null hypothesis H₁ : D > 0 using a paired t-test with the 16 differences. Let = 0.05.

Golfer	Brand A Distance	Brand B Distance
1	265	252
2	272	276
3	246	243
4	260	246
5	274	275
6	263	246
7	255	244
8	258	245
9	276	259
10	274	260
11	274	267
12	269	267
13	244	251
14	212	222
15	235	235
16	254	255

14. A company is comparing a new adhesive to an old adhesive. Let X and Y denote the amount of pressure the new and old adhesives can withstand, respectively.

(a) Based on n = m = 24 observations from both adhesives, define the test statistic and critical region for testing H₀ : _X _Y = 0 against H₁ : _X _Y > 0 where = 0.05
(b) For this part you will want to use the data in the files week7-X.txt and week7-Y.txt to calculate the value of the test statistic and state your conclusion.
(c) What is the approximate p-value of this test?
(d) You'll need to construct two box plots on the same graph. Do the box plots support your conclusion?

The next problem deals with a topic called Monte Carlo simulations.

In terms of machine learning, the phrase Monte Carlo generally means to run random simulations to estimate probability. We'll use it below to estimate the value of an integral.

Monte Carlo simulations (and the Monte Carlo Tree Search Algorithm) were used by Google's company DeepMind to help train a neural network in creating the computer programs AlphaGo and AlphaZero, which have become the leading players of the games of chess, shogi, and go. Their creation of AlphaGo "is the first computer program to defeat a professional human Go player, the first to defeat a Go world champion, and is arguably the strongest Go player in history."

15. Use Python (and NumPy) for this particular problem. Consider a function f(x) which is continuous on the interval [a, b]. Assume that m f (x) M for every x [a, b] for some constants m and M. We've already seen how we can estimate the integral

abf(x)dx

using Riemann sums, and other methods like the Trapezoid Rule, Simpson's Method, and Taylor series. We are going to see how randomly choosing points (x,y) can estimate the integral too! Consider the picture below.