These Hints Found Here In The Homework Market Site Might Hel

These Hints Found Here In the Homework Market Site Might Helpthe Ques

These hints found here in the Homework Market site might help. The questions, and dataset are in the attached file. Thank you.

Paper For Above instruction

Understanding the behavior of stock prices is crucial in financial analysis, and statistical tools such as probability, normal distribution, and quartiles provide valuable insights into stock market data. This paper explores key statistical concepts applied to Google stock prices over the past year, utilizing the Empirical Rule, standard normal distribution calculations, quartiles, and histogram analysis to evaluate stock price behavior and distribution characteristics.

Introduction

Stock prices exhibit variation over time, often assumed to follow certain probabilistic distributions. Analyzing historical stock data involves calculating measures like mean and standard deviation, exploring distribution properties, and determining the probabilities of certain events or anomalies. This analysis leverages concepts of the normal distribution, z-scores, quartiles, and data visualization techniques to assess whether a specific stock price movement is typical or unusual, and to understand the underlying data distribution.

Application of the Empirical Rule to Stock Prices

The Empirical Rule states that for a roughly normally distributed dataset, approximately 68% of data falls within one standard deviation of the mean, about 95% within two standard deviations, and approximately 99.7% within three standard deviations. By applying this rule to Google stock prices, we can estimate the probability that the stock closed at a price less than the mean. Due to the symmetry of the normal curve, the probability that the closing price was less than the mean is 50%, assuming symmetry (property #2 on p. 236). Therefore, the probability that on any given day the closing price was less than the mean is approximately 50%, regardless of the actual data, provided the distribution is symmetric and approximately normal.

Calculating Probability of Price Exceeding $500

To find the probability that the stock closed above $500, the process involves calculating the mean and standard deviation of the stock prices using Excel functions such as =AVERAGE() and =STDEV.S(). Once these are identified, the z-score for $500 (z1) is computed as:

z1 = (500 – mean) / standard deviation

Using the standard normal distribution table or Excel's =NORM.S.DIST(z1, TRUE), we find the cumulative probability P(z

P(z > z1) = 1 – P(z

This calculation determines the likelihood that on any given day, the stock closed at a price exceeding $500. If the probabilities are very low, exceeding $500 might be considered a rare or unusual event.

Assessing the Stock Price within a Certain Range of the Mean

To determine the probability that the stock closed within $45 of the mean, two z-scores are calculated: one for (mean + $45) and another for (mean – $45). These are given by:

z_upper = (mean + 45 – mean) / standard deviation = 45 / standard deviation

z_lower = (mean – 45 – mean) / standard deviation = –45 / standard deviation

Using these z-scores, the corresponding cumulative probabilities are obtained from the standard normal table or Excel functions. The probability that the stock price was within $45 of the mean is then the difference between these two cumulative probabilities:

P(|X – mean|

This approach helps quantify the likelihood of stock prices staying close to the average, indicating typical versus atypical price movements.

Determining Unusual Prices Using the Empirical Rule

A stock price claim at $700 per share can be evaluated for unusualness by calculating its z-score relative to the mean and standard deviation. Using the Empirical Rule, data points with z-scores exceeding 3 or less than –3 are often considered unusual. The z-score is computed as:

z = (700 – mean) / standard deviation

If the magnitude of z exceeds 3, the price at $700 is deemed unusual because it lies beyond three standard deviations from the mean, consistent with the properties of the normal distribution.

Identifying Statistically Unusual Closing Prices

To find the price bounds for unusual stock dynamics, we apply the Empirical Rule to define the upper and lower bounds as:

Lower bound = mean – 3 × standard deviation

Upper bound = mean + 3 × standard deviation

Any closing price outside these bounds may be considered statistically unusual. Calculating the standard deviation precisely from the dataset enables identifying these thresholds, indicating days with extraordinary price movements.

Calculating Quartiles of the Stock Price Data

Quartiles describe the distribution spread, with Q1, Q2 (median), and Q3 splitting the data into four parts. Using Excel functions such as =QUARTILE(array, 1), =QUARTILE(array, 2), and =QUARTILE(array, 3) to compute Q1, Q2, and Q3 allows for detailed distribution analysis. These values offer insights into the data’s skewness, spread, and central tendency, helping evaluate whether the stock data is symmetrically distributed or skewed.

Evaluating Normality of the Distribution

Assessing whether the data approximates a normal distribution involves constructing a histogram to visually inspect the shape. A bell-shaped histogram with symmetric sides suggests normality. Moreover, comparing mean and median indicates symmetry; if these are close, the distribution may be normal. Analyzing the differences among the quartiles, specifically whether Q2’s distance from Q1 and Q3 are approximately equal, further substantiates the normality assumption. These evaluations clarify whether statistical methods based on normality are appropriate for the dataset.

Conclusion

The statistical analysis of Google stock prices using probability models, the Empirical Rule, z-scores, quartiles, and histograms reveals the nature of stock price fluctuations. Assuming the data is approximately normal, certain prices are typical, while prices beyond three standard deviations are statistically unusual. Understanding these properties aids investors in identifying significant anomalies or normal market behavior, thereby informing better trading decisions. Nonetheless, the normality assumption should be validated through visual and numerical analysis, as real-world stock data may deviate from idealized models.

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