20 Jun stats-assignment
Theory
1 A statistical hypothesis is
T F T F T Fa random variable a pivotal quantity
an assertion about unknown distribution of one or more random variables
2 ∼Given arandom variable(absolutely)continue X F(x),F(x)unknown,siaX1,…,Xna sample from XandFn(x)the empirical distribution function.The Kolmogorov-Smirnovtest is usedtoverify
T F T F T Fthe hypothesis H0 : X ∼ F0 vs Ha : H0 is false, where F0(x) is the hypothesized distribution for X the hypothesis H0 : X hasthe Kolmogorov distribution vs Ha : X hasthe Smirnov distribution the hypothesis H0 : X ∼ F0 vs Ha : H0 is false, using the test statistic sup |Fn(x) − F0(x)|
3 x ∈ R TV F TV F TV F TV FIna hypothesis testthep−valueisthe significance 1−γ
the value of test statistic
the probabilityof rejecting the true hypothesis
the smallest significancewith which you couldreject H0 when it’s true
4 TV F TV F TV FIn a hypothesis testthesignificance level αisthe value of test statistic
the probabilitywith which you could reject H0 when H0 is true
the probabilitywith which you could accept H0 when H0 is false
5
),For a given experimentare only and only possible,sincompatible resultsSi(i=1,…,sand s≥2), each one of hypothesized probabilityπi=P[Si]; .Repeating n times the experiment the result Si will be observed Ni times. Obviously . Let ni the observed value of Ni and let the chi-squared with s-1 degrees of freedom. Answer correctly.
TV Fthe values represent the observed frequencies of the result Si
TV Fthe statistic is approximately normal when n is big
TV
F
TV Fthe condition is sufficient to ensure that
pag. 8
6 Given two hypothesis test simple,T1andT2to decide between the null hypothesis Hoand the alternativeHa,with error probability of 1stand 2ndtype respectively equal to α1, β1 for T1 and α2, β2 forT2, then T1 will be preferable to T2 if:
TV F TV F TV F TV Fα1 <α2 e β1 <β2 α1 = α2 e β1 >β2 α1 = α2 e β1 <β2 α1 >α2 e β1 <β2 Part B 1. Let U1 and U2 twouniform random variables on [0; 1] and independent. 1.1 To find the joint densityfU(u1,u2)of the vectorU=(U1,U2)Tand to draw the graph. 1.2 LetT=U1+U2.We denote by FT(t)the cumulative function of T.To show that, if 0≤t≤1,then FT(t)is the volume of parallelepiped in blue drawn in the picture. Calculate this volume. Hint:Calculate P[U1+U2≤t]whereas the points(u1,u2)favorable are such that {(u1,u2):0≤u1≤1;0≤u2≤1:0≤ u1 + u2 ≤t}. 1.3 Now, it will be easy to show that: Calculate the probability that Let now 0.523 0.556 0.994 1.000 1.052 1.087 1.154 1.322 1.485 1.549 (1) 10 observations from an unknown population X. 1.4 To write the sample distribution function Fn(x)concerning the observations(1). We want to decide about the test (non parametric) H0:FX(x)=FT(x) Ha : FX(x) Using the techniques of Kolmogorov-Smirnov. ≠ FT(x) (2) 1.5 At this scope,calledxk; withk=1,...,10the observations,we need first of all to complete the following table. 0.194 |FT (x6) − Fn(x5)| 0.002 |FT (x10) − 1| 1.6 Which of the values in the table is used to continue the test? 1.7 Using the Kolmogorov tables of the random variable(to see at the end of this assignment),todecide about the test(2)at a significance level α=0.05. 2 The company Startrip declares that the distance that his new Enterprise99 spaceship go through with 1 cubic meter (m3) of liquid hydrogen is normally distributed with mean μ0 higher than or equal to 20 parsecs (3.261563777 light years), and does not communicate the variance. The New NASA, to verify the Startrip declaration, shall obtain a preview of 5 different models of this rare ship model and test them just outside the Milky Way, noting the following average distance traveled (in parsecs) with 1 m3 of liquid hydrogen: 15.7 18.6 20.1 21.5 19.1. 2.1 Using these results you need to verify the Starship declaration.H0:µ≥µ0=20vs the alternativeHa:µ<µ0,with a level of significance α=1−γ=0.05. At the time of the Enterprise99 official presentation, Startrip announced that the mean μ0 of parsecs traveled with 1 m3 of hydrogen is exactly 20 with variance σ2 = 4 parsec2. Captain Kirk, who is interested to purchase an Enterprise99, heard that, on average, these ships travel 15 parsecs per m3 of hydrogen. He decides to buy an Enterprise99 (to accept the hypothesis H0: μ = μ0 = 20 vs the alternative H1: μ = 15) if the mean of parsecs traveled with 1 m3 of hydrogen from the first 50 Enterprise99 purchased by the Organization Intergalactic, exceeds 18. 2.2 If is the mean of the distances traveled by 50models purchased by Intergalactic Organization, indicatewhat are the regions of acceptance and rejection (i.e., where it’s need to drop the observed sample meanto accept or reject H0)if this is the strategy. 2.3 Calculate the probabilityαof the errorofthe 1st typeand βof the2ndtypeif this is thestrategy. 2.4 If x50=18.7what error can commit Captain Kirk? 3. William Sealy Gosset, tired of working for Guinness, where must publish its statistical work secretly under the pseudonym of Student, decided to quit and start to sell machinery to fill bottles of beer. Its first customer is obviously the same Guinness, which offers two different machines, M1 and M2, which fill the bottles of beer with an average content of 33 cl, but they do it with an accuracy that may not be the same. Guinness, having to buy one of the two machines, compares the performance of M1 and M2. Let X the quantity of beer that M1 releases in a bottle and using 61 bottles for the test, we obtain the following results: Xx61 =33.04cl, s2 = 0.0862 cl2, nX =61 Let Y the quantity of beer that M2 releases in a bottleand using 41 bottles for the test, we obtain the following results: Yy41 =33.07cl, s2 = 0.0885 cl2, nY =41 Where usually represent mean and sampling variances observed. Given the results of the observations, Guinness bets that M1 be more precise than M2 (i.e. ) 3.1 Wishing to verify with a test this bet, Guinnessprepares a parametric test, taking as alternative hypothesis . Therefore, indicate precisely the null hypothesis and the test statistic writing the formulas in term of variances of X and Y, its distribution (motivating the choice) and the relative degrees of freedom. 3.2 Indicate the critical region of level α=1−γ,and the type of test. 3.3 Evaluate the p-value of the data for the introduced test and derive a conclusion(hint:F0.75,40,60= 1.22). 3.4 The conclusion is strong or weak and why? Guinness would acquire so M1. But Gosset, which is astatistical expert, argues that M1 and M2 have the same accuracy. Therefore, it suggests to the Guinness to use the data as before for another test, where you takeas null hypothesis the fact that M1 and M2 have the same precision (i.e. ). . 3.5 Set, in this case, the statistical test, indicating the critical region of levelα=1−γ. 3.6 Evaluate the p-value of the data for the introduced test and derive a conclusion (hint: can be useful the value F0.5,60,40 = 1.0056). 3.7 The conclusion is strong or weak and why? 3.8 Considering that M1costs more than M2,which machine is convenient for Guinness?
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Fill in all the assignment paper details that are required in the order form with the standard information being the page count, deadline, academic level and type of paper. It is advisable to have this information at hand so that you can quickly fill in the necessary information needed in the form for the essay writer to be immediately assigned to your writing project. Make payment for the custom essay order to enable us to assign a suitable writer to your order. Payments are made through Paypal on a secured billing page. Finally, sit back and relax.
About Writedemy
We are a professional paper writing website. If you have searched a question and bumped into our website just know you are in the right place to get help in your coursework. We offer HIGH QUALITY & PLAGIARISM FREE Papers.
How It Works
To make an Order you only need to click on “Order Now” and we will direct you to our Order Page. Fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Are there Discounts?
All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.