1 (20 focuses) A buyer promotion bunch needs to decide if a soda pop maker offers at any rate 12 ounces of pop in a 12-ounce can. An irregular example of 10 jars is chosen from a conveyance van, and the measure of pop in each can is estimated. The information produced by the examination is given underneath: 12.07 12.04 12.05 11.02 12.05 11.35 12.01 12.00 11.95 12.15 Using ? = .05, would you be able to presume that the middle measure of pop in all jars is at any rate 12 ounces.

2. (20 focuses) Before a political race for the overall get together, 200 haphazardly chose up-and-comers were gotten some information about their democratic inclinations. 95 electors supported the Grassroots Party, 85 citizens supported Progressive Party, and the leftover 20 didn't any assessment. Would you be able to presume that electors favor one gathering over the other?

3. (20 focuses) Bill intends to put his home available to be purchased. Nonetheless, he needs to appraise the value he may get from the deal. He gathered information on home costs, land region, size of the home in square feet, and the age of the home, all in a tantamount area. The table underneath gives the information. Cost of the House (in great many dollars) Land Area (in sections of land) Size of the House (Square feet) Age of the House (a long time) 455 1.4 2500 8 278 .9 2250 12 463 1.8 2900 5 327 .7 1800 9 505 2.6 3200 4 264 1.2 2400 28 445 2.1 2700 9 346 1. 2050 13 487 2.8 2850 7 289 1.6 2400 16 434 3.2 2600 5 411 1.7 2300 8 223 .5 1700 19 a. Compose the populace relapse show and demonstrate the whether you anticipate a positive or negative connection between the reliant variable and every one of the autonomous factors. b. Utilizing Minitab, discover the relapse of cost ashore territory, house size, and age. c. Compose the assessed relapse condition. d. Which level of the all out variety in house costs is clarified by the free factors? e. Develop a 95 percent certainty span for the coefficient old enough. f. "Do " autonomous factors fundamentally influence the house cost at a 5% degree of importance? Offer purposes behind your response. g. Bill's 2000 square feet house sits on a 2-section of land and is 10 years of age. What amount would he be able to anticipate from the offer of his home?

4. (20 focuses) a. In an investigation of variable, what do the amount of squares inside (SSW) and the amount of squares between (SSB or SSG) mean. - Explain. How can it be that a single direction investigation of fluctuation is consistently an upper-tail test? b. What are the disservices of an unweighted value list? How is dealt with keep away from these drawbacks? c. What are the segments of a period arrangement? Give a short account on every one of these segments.

5. (20 focuses) The accompanying information shows costs of three kinds of natural products more than ten years. Year Mango Orange Blackberry 1 50 100 150 2 55 115 155 3 60 100 125 4 55 110 130 5 63 120 6 65 110 125 7 70 105 140 8 75 120 170 9 60 130 160 10 65 135 175 Take year 1 as the base year. Around there, Mango, Orange, and Blackberry creation were 50, 25, and 15 thousand kilograms, individually. a) Find the un-weighted record of costs. b) Find the Laspeyres list at natural product costs. c) What is the adjustment of cost from year 6 to year 7 in the event that you utilize an unweighted record. d) What is the adjustment of cost from year 6 to year 7 in the event that you utilize the Laspeyres file. e) Is the response to c is unique in relation to the response to d. Clarify your answer.

6. (20 focuses) Three providers give 500 one-gram parcels of saffron per shipment to a departmental store. Arbitrary examples of six shipments from every one of the three providers - Shujon Traders, Lavender, Just Quality - were painstakingly checked, and the quantity of bundles not adjusting to the quality particulars is tallied. These numbers are recorded in the table underneath. Shujon Traders Lavender Just Quality 28 22 33 37 27 29 34 29 39 29 20 33 31 18 37 33 30 38 At .05 degree of importance, test the speculation that populace mean a few parcels for each shipment not gathering the quality determination are something similar for every one of the three providers.

f(x | 0) = f(x | 1, 02) = (ex - 1)2 exp for r E R. 2TO 202 On the other hand, a Bayesian way of handling an unknown parameter (that is not of interest) is to integrate it out from the joint posterior density function. For a Bayesian inference on the vdB's problem, we assume a non-informative joint prior density for / and o' as follows to express our lack of knowledge about these two parameters: f ( 1, 0 ' ) x - 2'(MER, 03)0), or more simply f(A, a? ) of (a) (4 points) Find the marginal posterior density function of /, i.e., f(u | x) = / f( 1, 0? |2) do?. RT1. Basic Methods (35 points) In Bayesian inference the beta distribution is often used as the prior distribution for observed data with a binomial distribution. The beta distribution is normalized using the beta function which is dened on the interval [0, 1]: B(,)=j{;lfn_1{1$)_ldi, as >0 Assuming or = 5, and fl = 3, solve the integral to ve decimal points using the following methods: (a) (By Hand) Exact analytical solution. (b) (By Hand) Basic (noncomposite) Trapezoidal rule. (c) (By Hand) Basic (noncomposite) Simpson's 1,13 rule. Problem 2: Suppose X1,X2, ...,Xn ~ N(u,02) independently. Let us assume that 0'2 is known. Let X : E1321 Xg. ( 1) Consider the interval I : [X 1.960/,X + 1.960/ m Prove that for any xed value of it, Put E I) = 95%. (2) Consider the Bayesian inference, where we assume the prior distribution of p. is N(o:,'y2). Find the posterior distribution of pa. What is the posterior distribution as 72 > 00? For this limiting posterior distribution, prove that for any xed value of X, P01 E I) : 95%. (3) Discuss the difference between the concluding statements in (1) and (2). 3. Let Y ~Gamma(oc = 3, A = 0.5). Let X|Y = y ~Poisson()\\ = y). (a) Find E[X|Y]. (b) Find E[X]. (c) Find Var(X). o Note: Y is a continuous randoni variable representing the rate of occurence for the discrete random variable, X conditional on Y = :9. So, although we are dealing with a discrete and continuous pair of random variables, all of the formulae covered in class about condi tional expectation and variance still hold. This type of problem will be useful for Bayesian inference in Stat 61