代写board game、代做R、R程序语言调试、symmetric代做

日期：2019-05-12 09:54

1. A random walk board game is played as follows. A rook starts on the bottom row

of a chessboard that is infinite in the left, right and upwards directions. At every

step the rook moves randomly one square up, down, left, or right, where each of

the four directions is equally likely, independent of all other moves. When the rook

falls of the bottom of the board, its x coordinate is noted (relative to the initial

position which is x = 0).

This game has been played 29172 times for you and the results collected in the file

values.txt.

(a) Apply two different methods to evaluate whether the x values recorded follow

a roughly normal distribution.

(b) Does the distribution appear to have heavy tails or light tails? Discuss any

features of the random walk game which in your view would tend to cause this

property.

(c) Does the distribution of x values appear symmetric from the data? What about

from the description of the game? If there appears to be any discrepancy here,

how do you resolve it?

2. A hundred fish were captured at random in a lake and their weights in grams noted.

The data is in the file fish.txt.

(a) Obtain a histogram of the data set. Describe the shape and modality of the

data, as well as any skew.

(b) Apply two standard methods to evaluate heuristically whether the data seem

to be close to normal (i.e., drawn from an approximately normal distribution).

Make sure to include any R input and plots generated in your write-up.

(c) We are interested the true mean weight μ of all the fish in the lake. Describe

the approximate sampling distribution of X, the sample mean, in terms of

properties of the lake’s fish population. Explain any assumptions underlying

this approximation.

(d) Now calculate an approximate 95% confidence interval for μ on the basis of the

data in fish.txt and the sampling distribution you described in (c). Discuss

whether or not your answer to (b) undermines the validity of the confidence

interval you have calculated.

(e) Can you think of any practical issues with how the fish were captured that

might call into question whether X is an unbiased estimator of the true mean μ?

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3. If you are waiting some number of turns for a success that happens with a known

probability p, the expected number of turns until success is exactly 1/p, as we know

from our study of geometric random variables. If you have already been waiting

some time with no success, the expected additional waiting time until success is still

1/p.

If, however, success does not happen with the same probability at each turn (i.e., if

the turns are not i.i.d.), it is possible that the longer you have already been waiting,

the longer you will typically to have to wait in addition. This question describes a

simple model that can produce such an e?ect.

Consider a bag that contains one blue ball and one red ball. Each turn, you take

out a ball chosen uniformly at random from the bag, and return it to the bag along

with another (new) ball of the same colour. So after n turns there will always be

2 + n balls in the bag.

(a) Give an exact expression for the probability that the first time you draw a red

ball from the bag is on turn number n. (You do not have to simplify this.)

(b) In the Week 6 Tutorial you learned how to carry out a loop in R in order to

perform a random experiment that depends on an unknown number of turns.

Use a similar method to write a loop that simulates the number of turns taken

until the first red ball is drawn from the bag. Make sure to show your full R

code for the loop.

(c) Now modify your loop code to simulate the number of turns taken until the first

red ball is drawn from the bag, but this time where the bag starts with 10001

blue balls and one red ball. Run this code five times (to do this, remember you

may put multiple statements all on one line separated by semicolons in order

to simplify matters). Also run the code from part (b) five times.

(d) Suppose that we start with one red ball and one blue ball in the bag. Does it

appear from part (c) that you would typically have to wait less additional time

for the first red ball from this starting position than if you have already waited

10000 turns without finding a red ball? Suggest a statistical testing method

that may be able to verify this proposition. (You do not need to carry out the

test, but make sure to engage with the issues around appropriate assumptions

for whichever test you choose.)

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4. Five men and six women were sampled uniformly at random from the population.

Their heights in centimetres were as follows:

Let μM and μW be the mean height of all men and women respectively in the

population. Assume that the true variance of the mens’ heights and the true variance

of the womens’ heights are equal.

(a) Formulate a test (at the 5% significance level) for the null hypothesis

H0 : μM = μW

against the alternative hypothesis

HA : μM 6= μW .

Explain all assumptions made for this test and why they are reasonable to

assume.

(b) Write down the general formula for the test statistic and its distribution under

the null hypothesis.

(c) Compute the observed value of the test statistic, obtain an exact p-value (using

R) and write your conclusion in a sentence, being careful how you phrase it.

(d) Finally, suppose that we do away with the equal-variance assumption. Perform

a Welch’s t-test (with the conservative choice for degrees-of-freedom), again at

the 5% significance level, for the hypothesis

H0 : μM = μW

against the alternative hypothesis

HA : μM 6= μW .

Compare the conclusion with that obtained earlier under the equal-variance

assumption and discuss any difference.

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