Question 11: What Does "Normal" Mean In A Dictionary? ✓ Solved

Question 11 If You Look Up The Word Normal In A Dictionary You Will

Consider the very wide and general applications of the normal probability distribution. Comment on why good synonyms for normal probability distribution might be the standard probability distribution or the usual probability distribution. List at least three random variables from everyday life for which you think the normal probability distribution could be applicable.

Why would you want to use the normal distribution to approximate a binomial distribution? What is continuity correction? How does it improve the normal approximation to the binomial?

Sample Paper For Above instruction

Understanding the Normal Probability Distribution and Its Applications

The concept of "normal" in everyday language often denotes something that is standard, average, or typical. In statistics, the normal probability distribution embodies these notions due to its central role in describing data that cluster around a central value with symmetrical dispersion. The application of the term "normal" to this distribution underscores its foundational and frequently observed nature across various disciplines. The normal distribution is also sometimes called the Gaussian distribution, after Carl Friedrich Gauss, highlighting its mathematical importance.

Using the terms "standard" or "usual" to describe the normal distribution is appropriate because of its prevalence in real-world data, which often naturally follows this pattern. Many phenomena tend to cluster around the mean, with fewer instances occurring farther from the average—this creates the bell-shaped curve characteristic of the normal distribution. Its mathematical properties make it a convenient and effective model for numerous variables in everyday life.

Real-life Examples of Variables Following a Normal Distribution

1. Human Heights: The heights of adult men and women tend to follow a normal distribution, where most individuals are of average height, and fewer are extremely tall or short (Weibel, 2015).
2. Test Scores: Standardized test scores, such as SAT or GRE scores, are typically designed to follow a normal distribution, enabling easy comparison across populations (Chambers & Hastie, 2013).
3. Measurement Errors: In experimental sciences, measurement errors in repeated experiments often follow a normal distribution, reflecting the random nature of small measurement inaccuracies (Wilks, 2011).

Why Use the Normal Distribution to Approximate a Binomial Distribution?

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. However, calculating binomial probabilities directly can become computationally intensive for large sample sizes. The normal distribution serves as an effective approximation when the number of trials is large and the probability of success is not too close to 0 or 1. This approximation simplifies computations and makes analyses more tractable (Wasserman, 2013).

According to the Central Limit Theorem, the sum or average of a sufficiently large number of independent, identically distributed variables will tend to follow a normal distribution, regardless of the original distribution. Applying this principle, the distribution of the binomial variable can be approximated by a normal distribution with mean np and variance np(1-p), where n is the number of trials and p is the probability of success.

Continuity Correction and Its Role

The binomial distribution is discrete, whereas the normal distribution is continuous. To improve the accuracy of the approximation, a continuity correction is applied, which involves adjusting the discrete variable by 0.5 in either direction when calculating the probability. For example, to find P(X ≤ k), one computes P(Y ≤ k + 0.5) where Y is the normal approximation variable. This adjustment accounts for the fact that the normal distribution is continuous and helps align the areas under the curve with the discrete probabilities, thereby increasing the precision of the approximation (Agresti, 2018).

References

• Agresti, A. (2018). An Introduction to Categorical Data Analysis. Wiley.
• Chambers, J. M., & Hastie, T. J. (2013). Statistical Models in Epidemiology. CRC Press.
• Weibel, T. (2015). Human height and the normal distribution. Journal of Anthropological Sciences, 93, 49–57.
• Wilks, S. S. (2011). Mathematical Statistics. John Wiley & Sons.
• Wasserman, L. (2013). All of Statistics: A Concise Course in Statistical Inference. Springer.