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E(Y) = μly o V (Y) = n The first result says the sample means Y is an unbiased estimator of the population mean (that we are taking the draws from), and the second one says as the sample size increases the variance of the sample means becomes smaller and smaller. In this exercise, we are going to see whether we can obtain similar statements for the sampling distribution of sample proportions. Here we are interested in knowing the proportion of people in the population with a certain characteristic. Each member of the population either has this characteristic or not. Now, suppose that the proportion of people with a certain characteristic is p in the population and we take a random sample of size n and then compute the proportion of people with this characteristic in the sample, which is denoted by p. a) Show that E (p) = p. (Hint: Suppose X is the number of people in the sample with the desired characteristic, how would you express p using X and the sample size n? What is the distribution of X?) b) Show that V (p) p(1-P) 71 c) Interpret the results in the previous two parts.

E(Y) = μly o V (Y) = n The first result says the sample means Y is an unbiased estimator of the population mean (that we are taking the draws from), and the second one says as the sample size increases the variance of the sample means becomes smaller and smaller. In this exercise, we are going to see whether we can obtain similar statements for the sampling distribution of sample proportions. Here we are interested in knowing the proportion of people in the population with a certain characteristic. Each member of the population either has this characteristic or not. Now, suppose that the proportion of people with a certain characteristic is p in the population and we take a random sample of size n and then compute the proportion of people with this characteristic in the sample, which is denoted by p. a) Show that E (p) = p. (Hint: Suppose X is the number of people in the sample with the desired characteristic, how would you express p using X and the sample size n? What is the distribution of X?) b) Show that V (p) p(1-P) 71 c) Interpret the results in the previous two parts.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter2: Exponential, Logarithmic, And Trigonometric Functions
Section2.CR: Chapter 2 Review
Problem 111CR: Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data...
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E(Y) = V(Y) = μly n The first result says the sample means Y is an unbiased estimator of the population mean (that we are taking the draws from), and the second one says as the sample size increases the variance of the sample means becomes smaller and smaller. In this exercise, we are going to see whether we can obtain similar statements for the sampling distribution of sample proportions. Here we are interested in knowing the proportion of people in the population with a certain characteristic. Each member of the population either has this characteristic or not. Now, supp that the proportion of people with a certain characteristic is p in the population and we take a random sample of size n and then compute the proportion of people with this characteristic in the sample, which is denoted by p. a) Show that E (p) = p. (Hint: Suppose X is the number of people in the sample with the desired characteristic, how would you express p using X and the sample size n? What is the distribution of X?) b) Show that V (p) p(1-p) = c) Interpret the results in the previous two parts.
Transcribed Image Text:E(Y) = V(Y) = μly n The first result says the sample means Y is an unbiased estimator of the population mean (that we are taking the draws from), and the second one says as the sample size increases the variance of the sample means becomes smaller and smaller. In this exercise, we are going to see whether we can obtain similar statements for the sampling distribution of sample proportions. Here we are interested in knowing the proportion of people in the population with a certain characteristic. Each member of the population either has this characteristic or not. Now, supp that the proportion of people with a certain characteristic is p in the population and we take a random sample of size n and then compute the proportion of people with this characteristic in the sample, which is denoted by p. a) Show that E (p) = p. (Hint: Suppose X is the number of people in the sample with the desired characteristic, how would you express p using X and the sample size n? What is the distribution of X?) b) Show that V (p) p(1-p) = c) Interpret the results in the previous two parts.
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