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Problem 1 The amount of fill dispensed by a bottling machine is normally distributed with � = 1 ounce and unknown mean �. Let {�',�), ⋯ ,�+} → �(�, 1) be a random sample from this population and denote � = ∑ �2 + 23' . (1). Assuming that {�', �), ⋯ , �+}is an i.i.d. sample from �(�, 1). Specify the sampling distribution of �. (2) Under the same assumption as in (1), specify the sampling distribution of �) = ∑ ( + 23' �2 − �)) (3). Assume that the above sample size is � = 16, what is �(|� − �| ≤ 0.3)? (4). Under the i.i.d. assumption and � = 13, what is �(�) ≤ 5)? (5). [Bonus Point] If � = 3, {�', �),�?} are correlated but identically distributed as �(�, 1). Assume further that ���(�2,�C) = 0.5|2DC| . Specify the sampling distribution of �. (6). [Bonus Point] Under the same assumption as in (5), what is �(|� − �| < 0.3)? Problem 2 Let {�',�), . . . , �+'} →2.2.F. �(�', �' )) and {�', �), . . . , �+H} →2.2.F. �(�), �) )). Denote � = ∑ �2 +I 23' /�', � = ∑ �2 +H 23' /�), �' ) = ∑ ( +I 23' �2 − �))/(�' − 1), and � = ∑ �2 +H 23' /�), �' ) = ∑ ( +H 23' �2 − �))/(�) − 1). (1). Assume that both variances �' ) and �) ) are known, specify the sampling distribution of � − �. (2). Assume that both variances �' ) and �) ) are unknown and equal, specify the sampling distribution of � − �. (3). We assumed that the two population variances are equal. In fact, we can use the sample information to check that assumption. We know from Definition 7.3 (of our textbook) that � = �' )/�' ) �) )/�) ) is a pivotal quantity that follows an F distribution with �' − 1 degrees freedom in numerator and �) − 1 degrees of freedom in the denominator. Use this pivotal quantity to derive a formula for a 100(1 − �)% two-sided confidence interval for �) )/�' ). (4). (This part has three sub-parts) The flow of water through soil depends on, among other things, the porosity (volume proportion of voids) of the soil which is assumed to be normally distributed. To compare two types of sandy soil, �' = 16 measurements are to be taken on the porosity of soil A and �) = 21 measurements are to be taken on soil B. Assume that the two sample variances �' ) = .01 and �) ) = .02. (4a). Use the sample information given above to construct a 95% two-sided confidence interval for �) )/�' ). Is there any evidence that the two population variances are equal.[Hint: make sure you use correct lower and upper quantiles of the F distribution] (4b).Find the probability that the difference between the sample means will be within .005 unit of the difference between the population means �' − �). [Hint: use the result in (4a) to decide whether a pooled variance should be used] (4c). [Optional] If � − � = 0.03. Constructed a 95% confidence interval for �' − �). [Hints: (1).use the result in (4a) to decide whether a pooled variance should be used. (2). The lower quantile can also be found from the F table by using the formula: �+,R,'DS = 1/�R,+,S] Problem 3 Let Y be an exponential distribution with mean �, �(�) = V 1 � �DX/Y, � > 0; 0, ��ℎ������. (1). Show that � = 2�/� has a �) distribution with 2 degrees of freedom. That is, � is a pivotal quantity. (2). For a given �, find the lower confidence limit of �. (3). Assume that {�', �), ⋯ , �+} is an i.i.d. sample the above exponential distribution. Find the MLE of � (denoted by �c). (4). [Optional] Find the Fisher Information of �. Use the Fisher Information to find the variance of the MLE of �, denoted by �c, obtained in (3). Is �c a consistent estimator of �? Justify your answer.

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