BME339-Spring2024-HW5

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Mechanical Engineering

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Apr 3, 2024

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BME 339 BIOSTATISTICS I HOMEWORK #5 submit a single pdf on Gradescope Q1 (20 points). We have a population of 𝑋 = {5,10,15,20} . Construct the sampling distribution of sample mean (𝑥̅) when we draw samples of size 3 without replacement and ignoring order. Then, prove that 𝜇 𝑥̄ = 𝜇 and 𝜎 𝑥̅ 2 = 𝜎 2 𝑛 × 𝑁−𝑛 𝑁−1 . Q2 (20 points). researchers found the mean sodium intake in men and women 60 years or older to be 2940 mg with a standard deviation of 1476 mg. Use these values for the mean and standard deviation of the U.S. population, consider the sampling distribution of the sample mean based on samples sizes of 75 drawn from women in this age group. a) What is the mean and standard error of the sampling distribution of the sample mean? b) What type of distribution is the sampling distribution of the sample mean following? Why? c) Provide the pdf (formular) for the sampling distribution of the sample mean. d) What is the probability that a random sample will have a mean between 2500 and 2900mg? Q3 (20 points). The calcium levels in men and women ages 60 years or older are summarized in the following table: Use these estimates as the mean and standard deviation for the U.S. populations for these age groups. If we take a random sample of 40 men and 35 women, what is the probability of obtaining a difference between sample means of 100 mg or more? Q4 (20 points). It is known that 35 percent of the members of a certain population suffer from one or more chronic diseases. What is the probability that in a sample of 200 subjects drawn at random from this population 80 or more will have at least one chronic disease? Q5 (20 points). From the results of a survey conducted by the U.S. Bureau of Labor Statistics, it was estimated that 21 percent of workers employed in the Northeast participated in health care benefits programs that included vision care. The percentage in the South was 13 percent. Assume these percentages are population parameters for the respective U.S. regions. Suppose we select a simple random sample of size 120 northeastern workers and an independent simple random sample of 130 southern workers. What is the probability that the difference between sample proportions will be between .04 and .20?
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