An insurance company is building a simulation model to improve the total processing time of arriving...

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An insurance company is building a simulation model to improve the total processing time of arriving claims. Below is a sample of 30 processing times (in days) of insurance claims and some descriptive statistics. 4.25 4.03 8.76 Sample of 30 Processing Times 10.13 4.21 5.28 5.24 7.01 4.02 4.83 4.13 4.92 4.72 5.15 4.16 6.41 6.20 10.49 5.64 7.21 4.23 Sample mean Sample variance Median 5.83 3.25 5.26 Skewness coefficient Minimum Maximum eº(4-x), -{00%* 5.75 5.54 4.23 8.75 4.71 6.81 4.65 7.53 5.85 (a) Is normal distribution a good fit for this sample? If yes, give two distinct reasons why. If no, give two distinct reasons why not. (b) Suppose you decided to fit the data with a distribution that has density f(x)= x24 x < 4 1.24 4.02 10.49 www. where 8 >0 is an unknown parameter. Find a closed-form expression for the maximum-likelihood estimator of @ for an arbitrary sample X₁, X₂, X. Justify your answer by identifying the maximum likelihood (or log-likelihood) function and its relevant properties. (c) Calculate the maximum likelihood estimator for the given sample of processing times. (d) Perform a chi-square test for the null hypothesis that the sample conforms to the distribution specified in parts (b)-(c). Do this using five class intervals with equal probability (according to the hypothesized distribution) and for a = 0.05. Show all your calculations and your conclusion. You can use Excel function CHISQ.INV(1-a; v) for significance level a and v degrees of freedom (or the table at the end of the textbook) to determine the critical value. (e) Approximate the p-value. State what this value represents and whether the hypothesized distribution is a good fit or not, explain why. An insurance company is building a simulation model to improve the total processing time of arriving claims. Below is a sample of 30 processing times (in days) of insurance claims and some descriptive statistics. 4.25 4.03 8.76 Sample of 30 Processing Times 10.13 4.21 5.28 5.24 7.01 4.02 4.83 4.13 4.92 4.72 5.15 4.16 6.41 6.20 10.49 5.64 7.21 4.23 Sample mean Sample variance Median 5.83 3.25 5.26 Skewness coefficient Minimum Maximum eº(4-x), -{00%* 5.75 5.54 4.23 8.75 4.71 6.81 4.65 7.53 5.85 (a) Is normal distribution a good fit for this sample? If yes, give two distinct reasons why. If no, give two distinct reasons why not. (b) Suppose you decided to fit the data with a distribution that has density f(x)= x24 x < 4 1.24 4.02 10.49 www. where 8 >0 is an unknown parameter. Find a closed-form expression for the maximum-likelihood estimator of @ for an arbitrary sample X₁, X₂, X. Justify your answer by identifying the maximum likelihood (or log-likelihood) function and its relevant properties. (c) Calculate the maximum likelihood estimator for the given sample of processing times. (d) Perform a chi-square test for the null hypothesis that the sample conforms to the distribution specified in parts (b)-(c). Do this using five class intervals with equal probability (according to the hypothesized distribution) and for a = 0.05. Show all your calculations and your conclusion. You can use Excel function CHISQ.INV(1-a; v) for significance level a and v degrees of freedom (or the table at the end of the textbook) to determine the critical value. (e) Approximate the p-value. State what this value represents and whether the hypothesized distribution is a good fit or not, explain why.