Normal Distribution for Sample Mean

Module 2: Discussion Forum: Normal Distribution and Central Limit Theorem

Instructions

After reading the module, participate and comment on the following.

1. Guiding question:After having done the readings and research on the normal distribution and central limit theorem, answer the following premise explaining in detail your approach: “Can the normal distribution be applied to find probabilities for the sample mean, when the sample size is less than 30?”
2. Support your answer with specific references according to your readings, using the latest edition of APA format.
3. Include an example, definition, or application of the concept in daily life or work environment.
4. Enrich the content of your classmates with information or examples that help the understanding of the concepts and practices of normal distribution and central limit theorem.

General Instructions for the Discussion Forum

1. Post your answer as established by your instructor on the course calendar. Your comments must be written in your own words. You can offer examples and quotes to support your proposals. Citations of other authors must be properly documented (author’s name, title, date, etc.).
2. Post your comments to the response of at least two (2) of your classmates on or before the day set by your instructor on the course calendar. Your reaction may be based on personal experiences, study material, or additional information obtained from the Online LibraryLinks to an external site.or others, and may include:
   • Some understanding received from what is published that synthesizes the information and offers new perspectives or suggestions.
   • The validation or rejection of the idea (supported by your experience or research).
   • New information that broadens, adds or contrasts perspectives (based on reading and evidence).
3. Remember that your work must be original and must not contain material copied from books or the internet. You must respect the intellectual property of the authors and not commit plagiarism.
4. Examine the criteria used to evaluate your assignment to find out how to get the highest grade for your work. The assignments are graded or evaluated through rubrics or by the distribution of points.
5. Before submitting your entry, read your message several times. This will ensure that it contains the exact information you want to communicate.

Remember to review the academic expectations for your submission.

Submission Instructions:

• Submit your initial discussion post by 11:59 PM Eastern on Wednesday.
• Contribute a minimum of 350 words for your initial post. It should include at least 2 academic sources, formatted and cited in APA.
• Respond to at least two of your classmates’ discussion posts by 11:59 PM Eastern on Sunday. Ask a question, and provide a different viewpoint.

Can the Normal Distribution be Applied to Find Probabilities for the Sample Mean When the Sample Size is Less than 30?

The application of the normal distribution to find probabilities for the sample mean when the sample size is less than 30 requires careful consideration of the assumptions underlying the Central Limit Theorem (CLT). According to the CLT, regardless of the distribution of the population, the distribution of the sample means will approximate a normal distribution as the sample size increases. However, for smaller sample sizes (typically less than 30), the validity of this approximation can be questionable.

Supporting Argument: While the CLT provides a powerful tool for approximating the distribution of sample means, it is important to note that the theorem has certain requirements for its application. One crucial assumption is that the sample size is sufficiently large. Although the rule of thumb often cited is that a sample size of 30 or greater is considered large enough for the CLT to hold, this is not a strict cutoff. The adequacy of the sample size depends on various factors, including the skewness and kurtosis of the population distribution.

For smaller sample sizes, especially when dealing with non-normally distributed populations, alternative methods may be more appropriate for inference about the population mean. For instance, if the population distribution is known to be approximately normal or if the sample size is small but the population standard deviation is known, the t-distribution can be used instead of the normal distribution to calculate probabilities for the sample mean. The t-distribution is more robust to deviations from normality and becomes indistinguishable from the standard normal distribution as the sample size increases.

Example: Suppose a company wants to estimate the average time spent by customers on their website per visit. They collect a sample of 20 customers and calculate the mean time spent. Due to the small sample size, the company cannot confidently assume that the sample mean follows a normal distribution. Instead, they use the t-distribution to construct a confidence interval or conduct hypothesis testing about the population mean.

Conclusion: In conclusion, while the normal distribution is commonly used for inference about the sample mean, its applicability when the sample size is less than 30 should be approached with caution. Depending on the characteristics of the population distribution and sample size, alternative methods such as the t-distribution may be more suitable for making statistical inferences. Understanding the assumptions and limitations of these methods is crucial for accurate and reliable statistical analysis.

References:

1. Devore, J. L., & Berk, K. N. (2018). Modern Mathematical Statistics with Applications (2nd ed.). Springer.
2. Montgomery, D. C., Peck, E. A., & Vining, G. G. (2015). Introduction to Linear Regression Analysis (5th ed.). Wiley.