MAT-243 - 2-4 Discussion- The Central Limit Theorem

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Jan 9, 2024

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2-4 Discussion: The Central Limit Theorem Use the link in the Jupyter Notebook activity to access your Python script. Once you have made your calculations, complete this discussion. The script will output answers to the questions given below. You must attach your Python script output as an HTML file and respond to the questions below. In this discussion, you will apply the central limit theorem and use principles of the Normal distribution to calculate probabilities. You will demonstrate two key parts of the central limit theorem:  The distribution of sample means is approximately Normally distributed (bell-shaped) as the sample size increases and we repeatedly draw these samples, regardless of the shape of the population distribution from which the samples are drawn.  The average of all sample means is equal to the population mean. In practice, the average of all sample means will closely approximate the population mean. You will generate a population data set representing total precipitation (TPCP) in tenths of a millimeter using Python's numpy module. The distribution of this data set will be skewed. This data set will be unique to each student, and therefore the answers will be unique as well. Run Step 1 in the Python script to generate your unique population data. In your initial post, address the following items: 1. In the Python script, you created a histogram for the dataset generated in Step 1. Check to make sure that this data distribution is skewed and included in your attachment. See Step 2 in the Python script. 2. What is the mean of the TPCP population data? See Step 3 in the Python script. 3. In the Python script, you selected a random sample with replacement, of size 50 (note that this is a sufficiently large

sample), from the TPCP population. What is the mean of your random sample? Does this sample mean closely approximate the TPCP population mean? See Step 4 in the Python script. 4. You also selected 1,000 random samples of size 50 and calculated the mean of each sample. Then you stored those means into a dataframe. Check to make sure the output of this step is in your attachment. See Step 5 in the Python script. 5. Review the plotted data distribution for these 1,000 means. Does this approximate a Normal distribution? Does this confirm the first part of the central limit theorem? Why or why not? See Step 6 in the Python script. 6. What is the "grand" mean and standard deviation of these 1,000 means? Does the grand mean closely approximate (on a relative basis) the mean of the original distribution? Does this confirm the second part of the central limit theorem? Why or why not? See Step 7 in the Python script. Remember to attach your Python output and respond to all questions in your initial and follow-up posts. Be sure to clearly communicate your ideas using appropriate terminology. 1. There are two types of Skewed, the left and the right skewed. The histogram for the dataset generated in step 1 is a Skewed right distribution because we can see that has a peak of high-frequency data on the left with a tail of low-frequency data on the right side.

2. The mean of the TPCP population data is : Population mean = 503.23 3. The mean of my random sample is 513.36 which is 10.13 more than the Population mean. Here is the output of the 1,000 random samples of size 50 and calculated the mean of each sample. Normal Distribution in graphical form as a bell curve. As we can see in the figure below the graph forms a bell curve and indeed approximates a Normal distribution. This confirms the first part of the central limit theorem because it states that when an infinite number of successive random samples are taken from a population, the sampling distribution of the means of those samples will become approximately normally distributed with mean and standard deviation as the sample size becomes larger, irrespective of the shape of the population distribution.

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6. The Population mean is 503.23 while the grand mean (mean of 1000 sample means) is 501.65 confirming the second part of the central limit theorem because as stated the average of the sample means and the standard deviations will equal the population mean and standard deviation. Recall that your distribution of 1,000 means is Normally distributed even though the population distribution was skewed and the grand mean closely approximates the population mean. In your follow-up posts to other students, review your peers' results and provide some analysis and interpretation. 1. Is their population distribution skewed? Is their distribution of 1,000 sample means approximately Normally distributed? Does this confirm the first part of the central limit theorem? Why or why not? 2. Does their grand mean closely approximate their population mean? Does this confirm the second part of the central limit theorem? Why or why not?

3. Based on this discussion activity, what have you learned about the central limit theorem? Hello Yes, their population distribution is skewed to the right. As we can see it has a peak of high- frequency data on the left with a tail of low-frequency data on the right side. Yes, their distribution of 1,000 sample means is Normally distributed because as we can the graph is formed like a bell-shaped thus confirming the first part of the Normal distribution because with most values are clustering around a central region, and as it becomes thinner off as they go further away from the center. Yes, their grand mean is 472.24 closely approximately to their population mean which is 469.09 with a difference of 3.15. This confirms the second part of the central limit theorem because as the sample size gets larger, the grand mean approaches the mean of the original distribution. Based on this discussion activity, I’ve learned that as we increase the sample size, we are going to have a frequency plot that looks very close to a normal distribution. Hello Your population distribution graph shows a skewed right distribution. Your distribution of 1000 sample means confirms the Normal distribution because your graph shows a bell shape as the sample size grows. Your grand mean is 488.19 which is close approximately to their population mean which is 490.79 confirming the central limit theorem because as the sample size gets larger, the grand mean approaches the mean of the original distribution. Based on this discussion activity, taking a larger sample size will give us more accurate results.

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Student Absences and Grades on the Final Exam Dr. V. noticed that the more frequently a student is late or absent from class the worse he or she performs on the final exam. He decided to investigate. Dr. V. collected a random sample of 22 students. This sample includes the number of times a student is absent and their grades on the final exam. The data can be found in the Excel file Assignment10.xlsx. Do not use any software that I did not assign. Question 1: Which or the two variables (times absent or grades on the final exam) is the independent variable and which is the dependent variable? Question 2: Using Microsoft Excel: SHOW YOUR WORK Times Late/Absent Final Exam Grade 1 9 84.00 2 2 92.50 3 12 52.00 4 5 87.00 5 11 75.00 6 24 45.00 7 7 39.00 8 10 46.00 9 19 63.00 10 2 65.00 11 2 98.00 12 17 24.50 13 8 58.00 14 20 69.50 15 55 49.00 16 23 68.50 17 6 70.00 18 4 75.50 19 2 85.00 20 7 97.00 21 2 97.00 22 2 93.50 23 0 93.50 24 2…
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Chapter 2, Section 3, Exercise 102 Use the 95% rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about 95% of the data values.A bell-shaped distribution with mean 240 and standard deviation 28.The interval is to .

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