MAST 20005 Statistics Summer 2024
Assignment 2
Please have your name, student number, and your tutor’s name clearly displayed on the first 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 final 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 sufficient 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 H 0 : µ = 10
against the alternative hypothesis H 1 : µ < 10, let the critical region be defined by C = {¯ x : ¯ x ≤ 8 } , where ¯ x 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 significance level of the test?
(c) What are the values of K (8) and K (6) (the values of the power function when 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 effect 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 ( µ X , σ2 X) and N ( µ Y , σ2 Y ), respectively. A random sample from N ( µ X , σ2 X) of size n = 25 yielded ¯ x = 35 . 83 and s 2 x = 23 . 81, while a random
sample from N ( µ Y , σ2 Y ) of size m = 29 yielded ¯ y = 31 . 51 and s 2 y = 33 . 76.
(a) Assume σ 2 X = σ 2 Y , test the null hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth.
(b) If σ 2 X = σ 2 Y , test the null hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth.
(c) Test H 0 : σ 2 X = σ 2 Y against H 1 : σ 2 X = σ 2 Y at the α = 0 . 05 significance level.
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 p 1 be the probability of marking a successful free throw for a particular player (Player A). Since p 1 = 0 . 7, 2Player A decided to take a special training in order to increase p 1 . After the training was completed, Player A made 117 free throws out of 150 attempts.
