Stat 226 Module 4 Online Section 1 New York Stock Exchange

Stat 226 Module 4 Online Section1 New York Stock Exchange

Suppose that the percentage returns for a given year for all stocks listed on the New York Stock Exchange follow a non-normal distribution with mean µ = 12.4 percent and a standard deviation of σ = 8.4 percent. (a) Can you find the probability that a single stock has an annual return less than 34 percent? Yes or no. (b) Consider drawing a random sample of n = 5 stocks and calculating the mean return of these stocks. What is the shape, mean, and standard error of the sampling distribution of the sample mean for n=5? Can you find the probability that the mean return of the five sampled stocks is less than 34 percent? (c) For a sample size n=30, what are the shape, mean, and standard error of the sampling distribution? Using the CLT, what are the probabilities that the mean return is less than 10 percent and between 11 and 14 percent? (d) The CLT is illustrated using sample sizes n=5, 15, and 30 derived from data with a population mean of 3.3 and a standard deviation of 2.74. What is the shape of the population distribution? Specify the sampling distribution characteristics for these sizes. (e) Following procedures similar to those in (b) and (c), how does the standard error and shape change as sample size increases? (f) An automobile insurance company sells policies where the profit Y has a mean of $210 and a standard deviation of $7000. With 10,000 policies, what is the distribution of the sample mean profit? What is the shape, mean, standard deviation, and the probabilities Ȳ 45 dollars? Compute the total profits if the sample mean profit is $45 or $300, and determine the 5th and 95th percentiles of Ȳ.

Paper For Above instruction

The statistical principles surrounding the analysis of stock returns and insurance profits hinge fundamentally on probability distributions, the Central Limit Theorem (CLT), and inferential statistics. Understanding the behavior of these variables, especially how sample sizes influence the distribution shape and parameters, is essential for making informed financial and risk management decisions.

Starting with the New York Stock Exchange (NYSE) returns, the given information states that the annual returns follow a non-normal distribution with a mean (μ) of 12.4% and a standard deviation (σ) of 8.4%. A primary question is whether it is possible to determine the probability that a single stock’s annual return falls below 34%. To find this probability, one would need to know the distribution shape or rules to approximate or compute directly. If the distribution is perfectly known, calculating this probability involves standardizing the value using the z-score formula: z = (X - μ) / σ. In this case, for X=34%, the z-score would be (34 - 12.4) / 8.4 ≈ 2.54. However, since the distribution is non-normal, reliance solely on normal approximation may not be appropriate unless justified by the CLT application for large samples. Therefore, without additional information on exact distribution shape or tail behavior, the precise probability cannot be accurately computed. Nonetheless, if the normal approximation is considered plausible, then the probability that a stock has an annual return less than 34% is approximately 0.9946, indicating that such an event is highly probable.

When analyzing the sampling distribution of the sample mean, the size of the sample plays a crucial role. For n=5, according to the CLT, the shape of the distribution tends toward normality as the sample size increases, but with only five observations, the distribution may still be non-normal. The mean of the sampling distribution remains equal to the population mean, which is 12.4%. The standard error, a measure of variability of the sample mean, is calculated as σ/√n, resulting in approximately 8.4%/√5 ≈ 3.76%. Using these parameters, one could estimate the probability that the mean return of five stocks is less than 34% by standardizing and applying the normal CDF, yielding approximately 0.99999, assuming normality.

For larger sample sizes, such as n=30, the CLT assures that the sampling distribution of the mean is approximately normal, regardless of the original distribution's shape. The mean remains at 12.4%, and the standard error decreases to about 8.4%/√30 ≈ 1.53%. This smaller standard error indicates less variability among the sample means. With the normal approximation, the probability that the mean return is less than 10% can be calculated as the z-score for x̄=10%, which is (10 - 12.4) / 1.53 ≈ -1.56. Using standard normal tables, the probability is approximately 0.0594. Similarly, the probability that the mean return is between 11% and 14% can be calculated as the difference between the probabilities for the two Z-scores corresponding to these bounds.

The CLT's utility is further illustrated through a real-world dataset of student responses, where the distribution of the number of boyfriends or girlfriends has a mean of 3.3 and a standard deviation of 2.74. As the sample size increases from 5 to 15 and then to 30, the shape of the sampling distribution of the mean becomes more symmetric and approximately normal, consistent with theoretical expectations. Larger n reduces the standard error, resulting in narrower distributions of the sample mean, which enables more precise estimation of the true population mean. These concepts are fundamental in inferential statistics, allowing practitioners to make probabilistic statements about population parameters based on sample data.

In the case of automobile insurance profits, the profit per policy (Y) has a mean of $210 and a standard deviation of $7000. With a large number of policies (10,000), the distribution of the sample mean profit (Ȳ) can be modeled using the CLT, which states that Ȳ is approximately normal with the same mean (μ=210) but with a reduced standard deviation (standard error). Calculations show that the standard error is $7000/√10000 = $70. To find the probability that the average profit is less than $0, we calculate the z-score: (0 - 210)/70 ≈ -3.00, corresponding to a probability of approximately 0.0013, or 0.13%. For the probability that Ȳ exceeds $45, the z-score is (45 - 210)/70 ≈ -2.36, corresponding to a probability of about 0.0091 (or 0.91%).

If the sample mean profit turns out to be $45, the total profit from the 10,000 policies would be 10,000 × $45 = $450,000. If Ȳ is $300, the total profit is 10,000 × $300 = $3,000,000. Furthermore, the 5th and 95th percentiles of Ȳ can be derived from the standard normal distribution. The 5th percentile corresponds to a z-score of approximately -1.64, resulting in a lower bound: 210 + (-1.64 × 70) ≈ 94.8 dollars, and the 95th percentile corresponds to a z-score of 1.64, giving an upper bound of approximately 325.2 dollars. These estimates provide bounds within which the true average profit per policy is likely to fall with 90% confidence.

References

• Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
• Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. Chapman & Hall/CRC.
• Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics, 4th Edition. W. W. Norton & Company.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
• Ross, S. M. (2014). Introduction to Probability and Statistics. Academic Press.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
• Wackerly, D. D., Mendenhall III, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications. Cengage Learning.