REALLY NEED HELP WITH QUESTIONS 10-15. PLEASE PROVIDE DETAILED EXPLANATIONS WITH STEP-BY-STEP DETAILS AND ANSWER ALL QUESTIONS... PLEASE HELP!!!

Note: week7-X.txt and week7-Y.txt is located at the end of question 15

10. Let Y be b(100, p). To test H₀ : p = 0.08 against H₁ : p < 0.08, we reject H₀ and accept H₁if and only if Y 6.

• (a) Determine the significance level of the test.
• (b) Find the probability of a Type II error if in fact p = 0.04.

11. It was claimed that the proportion of Americans who select jogging as their main recreational activity is p = 0.25. A shoe manufacturer thought that p > 0.25, so they decided to test the following hypotheses:

H₀ : p = 0.25 and H₁ : p > 0.25. They asked 5757 Americans if jogging is their main recreational activity, and 1497 said yes.

• (a) Find the p-value for this test.
• (b) What is the manufacturers conclusion based on a significance level of = 0.05?

12. Assume that IQ scores for UIC graduate students are approximately N(, 100). To test

H₀ : = 110 against the one-sided alternative hypothesis H₁ : > 110, we take a random sample of size 16 from this population and observe that x = 113.5.

• (a) Do we accept or reject H₀ at the 10% significance level?
• (b) Do we accept or reject H₀ at the 5% significance level?
• (c) What is the p-value of this test?

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."

Here is a nice explanation of how Monte Carlo is used to create a program to play tic tac toe.

Here is how Monte Carlo simulations were used in finance.

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.