STA 220 MiniProject 2_Part 2_Group

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Jefferson Community and Technical College *

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220

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Statistics

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Apr 3, 2024

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Names: ______________________________________________________ 25 points possible STA 220 Mini-Project 2 – PART II Group Activity: The Central Limit Theorem Adapted from John Daniels – Central Michigan University PART II: Work in your group to answer the following questions based on your individual results from PART I. All members in the group are expected to collaborate to arrive at the answers. One student in each group needs to serve as the typist. The typist will download this document, type in answers below each question, save it, and then submit it. Before you begin, watch the first 7 minutes and 10 seconds of this video: https://youtu.be/NTq0BO6pbbg?si=kFFDRjV-nq7ABd9D Recall the premise of the Central Limit Theorem: The distribution of means of random samples of size n will approximately follow a normal distribution with mean μ x = μ and standard deviation σ x = σ √ n regardless the distribution of the population. The theory requires a sample size of at least 30 if the population distribution is unknown. However, because we know the distribution of rolls of a die is symmetric, we can get away with a much smaller sample size ( n = 4) and still see how the Central Limit Theorem works. We will now compare the results of rolling one die versus the experiment you performed: the mean of 4 rolls of a die. (Each question is worth 5 points.) 1. Comment on the difference in shape between the first and second histograms. 2. Do you believe that there is a Central Limit Theorem effect working with regard to the shape of the second distribution? Explain. 1

3. If the population of individual die rolls has a standard deviation of σ and the means of random samples of size n have a standard deviation of σ √ n (according to the Central Limit Theorem), which one of these two values will be lower if n>1? Hence, which one of these two distributions do you expect to have lower variability: the distribution of individual die rolls or the distribution of means of random samples taken from this population? Explain. 4. Looking at the first and second histograms, which distribution appears to have less variance: the distribution of individual die rolls (the population) or the distribution of sample means of size n = 4? Explain. Note: On Questions 3 and 4, by “distribution of individual die rolls (the population)” we mean how are the outcomes 1 through 6 distributed? We are not talking about the distribution or variability of the probabilities of the outcomes. 5. Do you believe that the Central Limit Theorem is working here with regards to the standard deviation of the sample means? Explain. 2

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STA 220 MiniProject 2_Part 2_Group

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