代写MAST 20005 Statistics Summer 2024 Assignment 2代做Java程序

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MAST 20005 Statistics Summer 2024

Assignment 2

Due: 11 pm, Tuesday 6 February 2024

Please have your name, student number, and your tutor’s name clearly displayed on the irst page.

Instructions: See the LMS for the full instructions, including the submission policy and how to submit your assignment. Remember to submit early and often: multiple submission are allowed, we will only mark your inal one. Late submissions will receive zero marks.

Questions labeled with‘(R)’require use of R. Please provide appropriate R commands and their output, along with su伍cient explanation and interpretation of the output to demon- strate your understanding. Such R output should be presented in an integrated form. together with your explanations; do not attach them as separate sheets. All other questions should be completed without reference to any R commands or output. Make sure you give enough explanation so your tutor can follow your reasoning if you happen to make a mistake. Please also try to be as succinct as possible. Each assignment will include marks for good presentation and for attempting all problems.

1. Assume that the distribution of X is N(μ, 25). To test the null hypothesis H0 : μ = 10 against the alternative hypothesis H1 : μ < 10, let the critical region be deined by C = f : 8g, where is the sample mean of a random sample of size n = 25 from

N(μ, 25).

(a) Find the power of this test as a function of the parameter μ, denoted as K(μ).

Hint: power is a function of the true parameter value.

(b) What is the signiicance level of the test?

μ = 8 and μ = 6)?

(d) (R) Sketch a graph of the power function. Hint: you may try μ from 4 to 12.

(e) What conclusion do you draw from the following 25 observations of X?

12.1 24.0 9.8 7.0 6.0 6.9 6.8 9.5

11.8 10.1 8.1 0.1 4.7 13.6 11.3 7.2 0.4

10.7 13.1 7.0 18.4 4.0 2.8 12.0 15.9

(f) What is the p-value of the test based on the observations in (e)?

2. Students looked at the efect of a certain fertilizer on plant growth. The students tested this fertilizer on one group of plants (Group A) and did not give fertilizer to a second group (Group B). Let X and Y denote the respective growths of the plants (in mm) in Group A and Group B over six weeks. Suppose X and Y are independent random variables with distributions N (μX , σX(2)) and N (μY , σY(2)), respectively. A random sample from N (μX , σX(2)) of size n = 25 yielded = 35.83 and sx(2) = 23.81, while a random sample from N (μY , σY(2)) of size m = 29 yielded = 31.51 and sy(2) = 33.76.

(a) Assume σX(2) = σY(2), test the null hypothesis at 1% signiicance level that the mean growths are equal against the alternative that the fertilizer enhanced growth.

(b) If σX(2) σY(2), test the null hypothesis at 1% signiicance level that the mean growths are equal against the alternative that the fertilizer enhanced growth.

3. In basketball, free throws or foul shots are unopposed attempts to score points by shooting from behind the free throw line (informally known as the foul line or the charity stripe), a line situated at the end of the restricted area. Free throws are generally awarded after a foul on the shooter by the opposing team, analogous to penalty shots in other team sports (from Wikipedia). Let p1 be the probability of marking a successful free throw for a particular player (Player A). Since p1 = 0.7,

Player A decided to take a special training in order to increase p1. After the training

was completed, Player A made 117 free throws out of 150 attempts.

(a) Test whether the training improved Player A’s free throw probability p1 or not,

that is, test H0 : p1 = 0.7 against H1 : p1 > 0.7 at α = 0.05 signiicance level.

(b) We want to compared Player A (after training) and another player (Player B) in terms of their free throw probabilities. Let p2 be the population proportion of successful free throw shots for Player B. For Player B, he made 109 free throws out of 128 attempts. Test H0 : p1 = p2 against the alternative hypothesis H1 : p1 < p2

at α = 0.05 signiicance level.

4. (R) Let X and Y be the scores in Probability (MAST20004) and Statistics (MAST20005),

respectively, for a student who took both of these two subjects. A sample of n = 15 students yielded the following data:

x y	x y	x y
54 56	65 73	73 71
62 64	81 74	85 81
61 71	63 59	88 82
75 66	86 86	90 84
77 75	68 66	58 59

(a) Fit a simple linear regression model E(YjX = x) = α + βx for these data. Find

point estimates of α and β . What are the standard errors of these estimates?

(b) Give a 95% conidence interval for the score in Statistics with a score of 70 in

Probability.

(c) Give a 95% prediction interval for the score in Statistics with a score of 70 in Probability. Compare the prediction interval to the conidence interval obtained

in (b).

(d) Use a t-test to test H0 : β = 0.8 against H1 : β 0.8 at the α = 0.05 signiicance

level and state your conclusion.

(e) Give the ANOVA table for the regression model.

(f) Use the ANOVA table (F-test) to test H0 : β = 0 against H1 : β 0 at the

α = 0.05 signiicance level and state your conclusion.

(g) Let ρ be the correlation coe伍cient between X and Y , ind an approximate 90%

conidence interval for ρ .

5. (R) A particular process puts a coating on a piece of glass so that it is sensitive to touch. Randomly throughout the day, 11 pieces of glass are selected from the production line and the resistance is measured on the glass. On each of three diferent days, December 6, December 7, and December 22, the following data give the measurements on each

of 11 pieces of glass:

December 6:

175.05

163.20

177.44

168.12

181.94

171.26

176.51

171.92

182.12

167.87

164.34

December 7:

175.93

174.67

176.62

174.27

171.39

177.16

173.90

184.13

178.34

167.21

172.90

December 22:

167.27

173.27

161.48

170.82

161.86

170.93

173.83

173.89

170.75

177.68

172.90

Perform a one-way analysis of variance to test whether the means on all three days are equal. State and test appropriate hypotheses at a 5% signiicance level. You should report the value of the appropriate statistic, the p-value, the assumptions you have made and your conclusions.

6. (R) In order to test whether four brands of gasoline give equal performance in terms of mileage, each of three cars was driven with each of the four brands of gasoline. Then each of the 3 4 = 12 possible combinations was repeated four times. The number of miles per gallon for each of the four repetitions in each cell is recorded in the following table:

Car

Brand of Gasoline

31.0

26.2

24.9

28.8

26.3

25.2

30.0

31.6

25.8

24.5

29.4

24.8

27.8

28.2

27.3

30.4

30.6

30.8

29.5

28.9

25.5

27.4

26.8

29.4

26.6

28.2

23.7

26.1

28.1

31.5

27.1

29.1

24.2

26.8

23.1

27.4

26.4

28.1

26.9

25.2

27.7

26.7

28.1

26.3

27.9

26.4

28.8

Perform. a two-way analysis of variance to examine whether these data suggest that the output is afected by the car and the brands of gasoline. State and test appropriate hypotheses at a 5% signiicance level. You should report the value of the appropriate statistic, the p-value, the assumptions you have made and your conclusions. 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. (R) When a stream is turbid, it is not completely clear due to suspended solids in the water. The higher the turbidity, the less clear is the water. A stream was studied on 26 days, half during dry weather (say, observations of X) and the other half immediately after a signiicant rainfall (say, observations of Y). The following turbidities were recorded in units of NTUs (nephelometric turbidity units):

1.9

2.1

10.9

1.7

5.5

3.8

11.6

2.4

8.4

6.1

6.6

6.2

2.6

8.8

5.4

5.2

5.1

2.2

9.4

13.9

10.5

8.2

23.4

11.4

10.7

6.5

(a) Using a signiicance level of 5%, perform an appropriate version of each of the following tests. In each case, state the null and alternative hypothesis.

i. two-sample t-test

ii. Wilcoxon two-sample test

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

8. The irrational number π = 3.1415926535.... is a mathematical constant. For i 2 f0, 1, 2, 3, 4, 5, 6, 7, 8, 9g, let pi be the proportion of decimal digits in π that equals i. One hypothesis on π is that the numbers 0, 1, . . . , 9 are equally (or discretely uniformly) distributed in the sense that pi = 1/10 for i = 0, 1, . . . , 9. We want to test this hypothesis statistically. It was counted that the frequencies of 0 - 9 in the irst 1000 decimal digits of π are

93, 116, 103, 103, 93, 97, 94, 95, 101, 105,

respectively. Would the data support the hypothesis that pi = 1/10 for i = 0, 1, . . . , 9?

Let α = 0.05.

9. Let X1 , . . . , Xn i.i.d. Uniform(0, θ), θ > 0.

(a) Show that the prior with pdf

π(θ) = { bab θ 0(-)b+1) , oth(θ) e,

is a conjugate prior for θ (a > 0 and b > 0 are known constants).

(b) Using this prior, calculate the posterior mean.