Pennies For Your Thoughts In Many Of The Problems From Last

Pennies For Your Thoughtsin Many Of The Problems From Last Uni

Topic: Pennies for Your Thoughts In many of the problems from last unit, you were given information about the population. For many of the variables, you assumed the variable had a normal distribution. What if the variables you are studying are not normally distributed? Here is your challenge – if the population is not normal, can you make any inferences about that population from your random samples?

As a class last unit, you have created a population of the ages of pennies. Your instructor will share this Penny Population document with you as an Excel file, that will include the histogram, mean, and standard deviation of this population of penny ages. Your instructor will also share a link to the Sampling Form to be used in step 5. See Example and DB starter video in Unit 5 LiveBinder.

Main Post

1) Describe the distribution shape of the population of penny ages (i.e., left skewed, symmetric, right skewed).

2) On your copy of the Penny Population document, randomly select 5 penny ages from this population. Calculate the mean of this Nickel Sample (sample size n = 5). How does this compare to the population mean?

3) On your copy of the Penny Population document, randomly select 10 penny ages from this population. Calculate the mean of this Dime Sample (sample size n = 10). How does this compare to the population mean?

4) On your copy of the Penny Population document, randomly select 25 penny ages from this population. Calculate the mean of this Quarter Sample (sample size n = 25). How does this compare to the population mean?

5) Enter your Nickel Sample mean, Dime Sample mean, and Quarter Sample mean in the Sampling Form sent by your instructor. It will generate a histogram of the class means for each sample (Nickel Samples, Dime Samples, and Quarter Samples).

6) Copy and paste the class histograms as they look so far for others to review in the discussion.

Peer Reply #1

Review a classmate’s post. Respond to them as a friend. In a few sentences, explain to them what the Central Limit Theorem says referencing the mean for their Nickel Sample, Dime Sample, and Quarter Sample.

Peer Reply #2

Review another classmates’ post. Respond to them as a friend. What do you notice about the shape of the histograms for the Nickel samples, Dime samples, and Quarter samples? Do any of their histograms look normal? What can you infer about a Half Dollar Sample (sample size n = 50)?

See Example. Activity based on R.L. Schaeffer et al., Activity-Based Statistics.

Paper For Above instruction

The investigation into the distribution of penny ages provides an insightful case study into how sample size influences our ability to make inferences about a population, especially when the underlying population distribution is not normal. This exploration underscores vital statistical concepts such as the distribution shape, the Central Limit Theorem, and the importance of sample size in sampling distributions.

Firstly, understanding the shape of the population distribution is fundamental. In many real-world scenarios, data such as penny ages may exhibit skewness rather than a perfect normal distribution. For instance, the age distribution of pennies could be right-skewed if most pennies are relatively recent while a few are very old, or left-skewed if older pennies are more common. Visual inspection of the histogram shared by the instructor can reveal whether the distribution is symmetric or skewed. Recognizing the shape is essential because many statistical inference techniques assume normality, which might not be valid if the population distribution is heavily skewed.

Secondly, the process of sampling and calculating the means from small samples (n=5, 10, 25) elucidates how sample size impacts the distribution of the sample mean. When selecting five pennies at random and computing the mean, small sample variability tends to be high, and the sample mean may deviate considerably from the population mean. This variability diminishes as the sample size increases—larger samples tend to produce means closer to the true population mean due to the Law of Large Numbers.

Specifically, the sampling distribution of the mean tends to be approximately normal as the sample size grows, even if the population distribution is skewed, as stated by the Central Limit Theorem (CLT). For the samples with n=5, the mean distribution may still exhibit skewness or irregularities, but with n=10, some approximation towards normality begins to appear. At n=25, the distribution of the sample means should resemble a normal distribution, providing a basis for making valid inferences about the population mean, despite the original skewed distribution.

Inserting the sample means into the Sampling Form and analyzing the resulting histograms reveals these trends visually. The histograms for the smaller samples (Nickel and Dime) might still display irregularities or skewness, but the histogram for the larger Quarter sample (n=25) should appear more symmetric and bell-shaped, exemplifying the CLT's implications. This demonstrates why larger samples are more reliable when trying to infer population parameters.

Regarding the shape of the histograms, the expectation is that as the sample size increases, the histogram of the sample means becomes more normal—even if the original population distribution is skewed. For smaller sample sizes, the distribution may retain skewness or irregularities, complicating assumptions of normality. The normal-looking distribution for the n=50 Sample (Half Dollar), which is even larger, should be narrower and more tightly clustered around the population mean, providing even more robust inferential power.

In conclusion, the study highlights several core statistical principles. The shape of the original population distribution significantly affects the sampling distribution of the mean for small samples. However, the Central Limit Theorem assures that with increasing sample size, the distribution of sample means approaches normal regardless of the original population shape. This property is crucial because it justifies the use of normal approximation techniques in inferential statistics, even when the population distribution is unknown or non-normal. Therefore, understanding sample size and distribution shape aids in making more accurate and reliable inferences about the population based on sample data.

References

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• Schaeffer, R. L., et al. (2018). Activity-Based Statistics. Pearson.
• Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
• Castillo, E. (2010). Introduction to Probability and Statistics. CRC Press.
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• Larson, R., & Farber, M. (2014). Elementary Statistics: Picturing the World. Pearson.
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