A, B and C are three circles of unit radius with centres in the x y-plane at (1,2),(2.5,1.5) and (2,3), respectively. Devise a hit or miss Monte Carlo calculation to determine the size of the area that lies outside C but inside $A$ and $B$, as well as inside the square centred on $(2,2.5)$, that has sides of length 2 parallel to the coordinate axes. You should choose your sampling region so as to make the estimation as efficient as possible. Take the random number distribution to be uniform on $(0,1)$ and determine the inequalities that have to be tested using the random numbers chosen.

Using the points and weights given in table , answer the following questions. (a) A table of unnormalised Hermite polynomials $H_n(x)$ has been spattered with ink blots and gives $H_5(x)$ as $32 x^5-? x^3+120 x$ and $H_4(x)$ as $? x^4-? x^2+12$, where the coefficients marked ? cannot be read. What should they read? (b) What is the value of the integral

$$I=\int_{-\infty}^{\infty} \frac{e^{-2 x^2}}{4 x^2+3 x+1} d x,$$

as given by a 7-point integration routine?

Use an iteration procedure to find the root of the equation $40 x=\exp x$ to four significant figures.

You must choose an SRS of 10 of the 440 retail outlets in New York that sell your company's products. How would you label this population to select a simple random sample?

(a) 001, 002, 003, . . . , 439, 440

(b) 000, 001, 002, . . . , 439, 440

(c) 1, 2, . . . ,439, 440

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A simple random sample of FICO credit rating scores was obtained, and the sorted scores are listed below. Make a boxplot and include the values of the 5-number summary.

$$\begin{array}{llllllllllll} 664 & 693 & 698 & 714 & 751 & 753 & 779 & 789 & 802 & 818 & 834 & 836 \end{array}$$

In 1970, 59% of college freshmen thought that capital punishment should be abolished; by 2005, the percentage had dropped to 35%. Is the difference real, or can it be explained by chance? You may assume that the percentages are based on two independent simple random samples, each of size 1,000.

A simple random sample of birth weights in the United States has a mean of 3433 g. The standard deviation of all birth weights is 495 g.

c. Which of the preceding confidence intervals is wider? Why?

The solid represented by the Mat Plan is built and glued together. The solid is dipped in red paint and then left to dry. Once dried, the cubes are broken apart and each cube is dropped into a bag. If you are going to reach into the bag and pull out a cube at random, what is the probability that the cube has at least one red face?

Use, the given confidence level and sample data to find a confidence interval for the population standard deviation $\sigma$. Assume that a simple random sample has been selected from a population that is a normal distribution.

95% confidence; $n=30, \bar{x}=1533, s=333$

Test the given claim. Assume that both samples are independent simple random samples from populations having normal distributions.

A random sample of 13 four-cylinder cars is obtained, and the braking distances are measured and found to have a mean of $137.5 \mathrm{ft}$ and a standard deviation of $5.8 \mathrm{ft}$. A random sample of 12 six-cylinder cars is obtained and the braking distances have a mean of $136.3 \mathrm{ft}$ and a standard deviation of $9.7 \mathrm{ft}$. Use a $0.05$ significance level to test the claim that braking distances of four-cylinder cars and braking distances of six-cylinder cars have the same standard deviation.

The inside diameter of a randomly selected piston ring is a random variable with mean value $12{cm}$ and standard deviation $.04 {cm}$. a. If $\bar{X}$ is the sample mean diameter for a random sample of $n=16$ rings, where is the sampling distribution of $\bar{X}$ centered, and what is the standard deviation of the $\bar{X}$ distribution? b. Answer the questions posed in part (a) for a sample size of $n=64$ rings. c. For which of the two random samples, the one of part (a) or the one of part (b), is $\bar{X}$ more likely to be within $.01 {cm}$ of $12 {cm}$ ? Explain your reasoning.

The number of dirt specks on a randomly selected square yard of polyethylene film of a certain type has a Poisson distribution with a mean value of 2 specks per square yard. Consider a random sample of $n=5$ film specimens, each having area 1 square yard, and let $\bar{X}$ be the resulting sample mean number of dirt specks. Obtain the first 21 probabilities in the $\bar{X}$ sampling distribution.

A piece of plywood is composed of five layers. The layers are a simple random sample from a population whose thickness has mean 0.125 in. and standard deviation 0.005 in. a. Find the mean thickness of a piece of plywood. b. Find the standard deviation of the thickness of a piece of plywood.