MAST20005: Mathematics Assignment

Problems:

1. (R) The data in the le icecream gives a measure of the consumption of an ice cream (1000 per day), the price (in dollars) of the ice cream and mean temperature (?C).

(a) Draw the scatter plots of the consumption versus the price and temperature, respec tively. Which covariate seem to have a stronger relationship with the consumption of the ice cream?

(b) Fit a simple linear regression model to the consumption and the covariate you chose in (a). Show an appropriate summary of the model t.

(c) Draw a standard set of diagnostic plots. What do you see?

(d) Give a two-sided 90% con dence interval for the consumption of the ice cream when the mean temperature is 30.

(e) A merchant wants to predict the consumption when the mean temperature is 25. Give a two-sided 95% prediction interval for that.

(f) Let?be the correlation coe cient between the ice cream consumption and the mean temperature, nd an approximate two-sided 95% con dence interval for?.

2. A statisticTbased on a random sample ofnobservations is such thatT ?Bi(n, p). It is required to useTto testH0:p= 0.2 againstH1:p >0.2.

(a) Consider a sample ofn= 50 observations and use the normal distribution approxi mation.

1. Determine the rejection region forTwith signi cance level 0.05.
2. Find the power of this test whenp= 0.3.

(b) Using the normal distribution approximation, find a sample size to testH0:p= 0.2 againstH1:p >0.2 with signi cance level 0.01 and power 0.99 whenp= 0.3.

3. (R) Consider a random sample from a continuous distributionX1, . . . , Xn. Assume that the distribution is normal andn= 20 and the observed data are:

-0.30, -1.06, -0.81, -0.53, -0.20, 1.00, 1.62, 0.40, -0.52, 1.03, 0.18, 1.11, -0.12, -0.19, 0.84, -1.36, -0.09, 0.48, 0.45, 0.51

We wish to test if the mean of the distribution is 0.5.

(a) Using a signi cance level of 5%, perform an appropriate version of each of the fol lowing tests. In each case, state the null and alternative hypotheses.

1. Wilcoxon test
2. t-test
3. sign test

(b) How do the conclusions of these tests compare with each other? Explain your answer and what conclusion you would form overall.

(c) Estimate (via simulation) the power of the above tests if the true distribution isX ?N(0.35,0.72).

4. A researcher wants to know if there is a relationship between smoking and exercise levels in women and collected the following data. Help the researcher to test if smoking and exercise levels in women are independent use a signi cance level of 0.05.

(a) State the null hypothesisH0, calculate the test statistic and give the distribution of the test statistic with the parameter underH0.

(b) Assuming the null hypothesis is true, what is the expected number of women whose exercise levels are between 200 - 599 and never smoke?

(c) (R) Conduct an appropriate test, give the p-value and draw your conclusions.

5. LetX1, . . . , XnandY1, . . . , Ymbe two independent random samples from N(X,52) and N(Y,52), respectively.

(a) To testH0:X= 100 againstH1:X>100 at a?signi cance level, what is the critical region speci ed by the likelihood ratio test criterion?

(b) Show that the likelihood ratio for testingH0:X=YagainH1:x?=Yis afunction of the usual two-sample statistic for known?, (x ? y)/(?p1/n+ 1/m).

6. (R) A psychology student was interested in testing how food consumption by rats would be a ected by a particular drug. She used two levels of one attribute, namely, drug and placebo, and four levels of a second attribute, namely, male (M), castrated (C), female (F), and ovariectomised (O). For each cell, she observed two rats. The amount of food consumed in grams per 24 hours is listed in the following table:

(a) Perform a two-way analysis of variance to examine whether these data suggest that food consumption by rats a ected by the drug. State and test appropriate hypotheses at a 5% signi cance level. You should report the value of the appropriate statistic, the p-value, the assumptions you have made and your conclusions.

(b) Is it possible to test for interaction? If yes, then perform the test and draw an interaction plot; otherwise, explain why it is not possible.

7. Consider a random sampleX1, . . . , Xnfrom a continuous distribution with the probabilitydensity function:

where? >0 is an unknown parameter. Given a random sample of sizen= 4: 1.2,2.4,1.9,1.5.

(a) 1. Find the maximum likelihood estimate of?.

2. Show thatW= (X(n)?1)/?is a pivot.

3. Construct a two-sided 95% con dence interval for?.

(b) 1. Find thep-quantile,?p, off(x|?) in terms of?.

2. Name the asymptotic distribution of the sample median,M, and give its mean and variance in terms of?.

3. Calculate the standard error ofM.

8. To study the state of a particular sh catchment, you measure the length of sh caught during a one-week period. Based on previous studies, you decide to model these mea surements using a normal distribution with a mean of?cm and a standard deviation of 4 cm. The data you obtain consists of 50 measurements, with a sample mean of 20 cm and sample standard deviation 3 cm. Use an improper uniform prior for?.

(a) What is an improper prior?

(b) Find the posterior distribution of?.

(c) The catchment would be considered healthy if the population mean is at least 20.5 cm. Based on the data and model above, give an approximate posterior prob ability that this catchment is healthy.

(d) Calculate the 95% credible upper bound for?.

Get your Mathematics assignment solved by our Mathematics Experts from Exam Question Bank . Our Assignment Writing Experts are efficient to provide a fresh solution to all question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing Style. Be it a used or new solution, the quality of the work submitted by our assignment experts remains unhampered.

You may continue to expect the same or even better quality with the used and new assignment solution files respectively. Theres one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turn tin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.