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Problem 1 A retail dealership sells three brands of automobiles. For brand 𝐴, the profit per sale, 𝑋 is normally distributed with parameters (πœ‡%, 𝜎% (); for brand 𝐡 the profit per sale π‘Œ is normally distributed with parameters (πœ‡(, 𝜎( (); for brand 𝐢, the profit per sale π‘Š is normally distributed with parameters (πœ‡., 𝜎. (). Assume that 𝑋, π‘Œ, and π‘Š are independent. For the year, two-fifths of the dealer’s sales are of brand A, one-fifth of brand 𝐡, and the remaining two-fifths of brand 𝐢. Let {𝑋%, 𝑋(,β‹―,𝑋23} 556 78𝑁(πœ‡%, 𝜎% (), {π‘Œ%, π‘Œ(,β‹―, π‘Œ2:} 556 78𝑁(πœ‡(, 𝜎( (), and {π‘Š%, π‘Š(,β‹―,π‘Š2;} 556 78𝑁(πœ‡., 𝜎. (). Define 𝑋 < = Ξ£ (𝑋5/𝑛% 23 ) 5A% , π‘Œ < = Ξ£23 (π‘Œ5/𝑛() 5A% , π‘Š B = Ξ£23 (π‘Š5/𝑛.) 5A% . 𝑆% ( = % 23D% Ξ£ (𝑋5 βˆ’π‘‹ < 23 )( 5A% , 𝑆( ( = % 2:D% Ξ£ (π‘Œ5 βˆ’π‘Œ < 2: )( 5A% , and 𝑆. ( = % 2;D% Ξ£ (π‘Š5 βˆ’π‘Š B 2; )( 5A% . The quantity π‘ˆ = .4𝑋 < + .2π‘Œ < + .4π‘Š B will approximate to the true average profit per sale for the year. (1). What is the distribution of U? Find the mean, variance, and probability density function for π‘ˆ. (2). [Bonus Problem] Derive the likelihood ratio statistic to test the following hypothesis regarding the above three normal population means and variances. Ho: πœ‡% = πœ‡( = πœ‡% = πœ‡ and 𝜎% ( = 𝜎( ( = 𝜎. ( = 𝜎( v.s. Ha: at least one of the above qualities does not hold (Ha says the three normal populations are not identical. That is, at least one of the three means is different from the others or at least one of the three variances is different from the other variances) The following steps direct you to derive the LR test statistic. (2a) Write the unrestricted likelihood function 𝐿(πœ‡%, 𝜎% (, πœ‡(, 𝜎( (, πœ‡., 𝜎. () and find the MLE of (πœ‡%, 𝜎% (, πœ‡(, 𝜎( (, πœ‡., 𝜎. (), denoted by (πœ‡Μ‚%, 𝜎M% (, πœ‡Μ‚(, 𝜎M( (, πœ‡Μ‚., 𝜎M. (). If you need additional notations, please list them before you use them. (2b). Find the restricted likelihood function 𝐿(πœ‡, 𝜎() under Ho: πœ‡% = πœ‡( = πœ‡% = πœ‡ and 𝜎% ( = 𝜎( ( = 𝜎. ( = 𝜎( and the restricted MLE of (πœ‡, 𝜎(), denoted by (πœ‡N, 𝜎N (). (2c). Find the explicit expression of the following LR test statistics in terms of the sample data and specify the sampling distribution of the test statistic. βˆ’2 log(πœ†) = βˆ’2 log 𝐿(πœ‡N, 𝜎N () 𝐿(πœ‡Μ‚%, 𝜎M% (, πœ‡Μ‚(, 𝜎M( (, πœ‡Μ‚., 𝜎M. () Problem 2 Let {𝑋%, 𝑋(,β‹―, 𝑋2} 556 78 𝑓(π‘₯) = 2πœƒπ‘₯𝑒DWX: (πœƒ > 0, π‘₯ > 0), where 𝑓(π‘₯) is Rayleigh distribution. 1. Find maximum likelihood estimator (MLE) of πœƒ, denoted by πœƒ[. 2. Find the Fisher information of πœƒ and show that MLE of πœƒ is a consistent estimator. Problem 3. The length of time between billing and receipt of payment was recorded for a random sample of 100 of a certified public accountant (CPA) firm’s clients. Let {𝑋%,𝑋(,β‹―,𝑋%\\} be a random sample taken from the population 𝑓(π‘₯) with sample mean 𝑋 < = Ξ£ 𝑋5 %\\ /100 5A% =39 and standard deviation 𝑆 = ^Ξ£ (𝑋5 βˆ’π‘‹ < %\\ )(/99 5A% = 35, respectively. Let πœ‡ be the population mean. Part I. Test the claim that the mean length of time (πœ‡) between billing and receipt of payment is less than 40 days at level 0.05 using the following methods. (1). Critical value method. (2). Confidence interval method. (3). P-value method. Part II. Likelihood ratio test. To use the likelihood ratio test, we assume that the waiting times are exponentially distributed with density function 𝑓(π‘₯) = b% cd 𝑒DX/c (πœ‡ > 0, π‘₯ > 0). That means random sample {𝑋%, 𝑋(,β‹―, 𝑋%\\} 556 78 𝑓(π‘₯) = b% cd 𝑒DX/c . As stated earlier, 𝑋 < = 39 and 𝑆 = 35. Test the following claim that the mean length of time (πœ‡) between billing and receipt of payment is equal to 40 days at level 0.05. Problem 4. This is not a traditional exam question. The idea is to summarize how to choose an appropriate inferential procedure for a given inferential problem based on the information available in both the sample and the population. As an example, I created two flow-charts that give detailed steps for making inferences about a population mean (πœ‡) or a population proportion (p). You can draw the flow-charts by hand. What you are expected to do: Similar to the above two charts concerning single population mean and proportion, please draw two charts in this problem based on the following given instructions. The first chart should be based on either 4.1. or 4.2 and the second chart should be based on either 4.3. or 4.4. 4.1. Draw a flow-chart to show the steps for constructing confidence intervals for the difference of two population proportions (p1-p2) and the difference of two population means (πœ‡% βˆ’ πœ‡() based on the given information in the data and the assumptions of the populations. 4.2. Draw a flow-chart to show the steps for testing hypotheses about the difference between two population proportions (p1-p2) and the difference of two population means (πœ‡% βˆ’ πœ‡() based on the given information in the data and the assumptions of the populations. 4.3. Draw a flow-chart to show the steps for constructing confidence intervals for a population variance (𝜎() and the ratio of two population variances (𝜎% ( 𝜎( ⁄ () based on the given information in the data and the assumptions of the populations. 4.4. Draw a flow-chart to show the steps for testing hypotheses about a population variance (𝜎() and the ratio of two population variances (𝜎% ( 𝜎( ⁄ () based on the given information in the data and the assumptions of the populations.

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