Assume that one has k independent random samples of sizes n1

Assume that one has k independent random samples of sizes n_1, n_2, n_3,..., n_k from the same distribution. These samples generate k unbiased estimators for the mean, namely Show that the arithmetic average of these estimators, is also unbiased for mu Certain mineral elements required by plants are classed as macronutrients. Macronutrients are measured in terms of their percentage of the dry weight of the plant. Proportions of each element vary in different species and in the same species grown under differing conditions. One macronutrient is sulfur. In a study of winter cress, a member of the mustard family, these data, based on three independent random samples, are obtained: Use the result of part (a) to obtain an unbiased estimate for mu, the mean proportion of sulfur by dry weight in winter cress. By averaging the three values .8, .95, and .7 to obtain the estimate for mu, each sample is being given equal importance or \"weight.\" Does this seem reasonable in this problem? Explain. To take sample sizes into account, a \"weighted\" mean is used. This estimator, is given by Show that is an unbiased estimator for mu. Use the data of part (b) to find the weighted estimate for the mean proportion of sulfur by dry weight in winter cress. Compare your answer to the estimate found in part (b).

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