Based On Larson Farber Sections 5153go Tohttpwwwgooglecomf
Based on Larson & Farber: sections 5.1–5.3. One must download the stock data for Google (GOOGL) spanning from November 20, 2012, to November 19, 2013, and analyze the closing prices as a normally distributed data set. The assignment involves calculating various probabilities related to the stock's closing prices, utilizing properties of the normal distribution, the empirical rule, and Excel functions. The tasks include determining the probability that a randomly selected closing price was less than the annual mean, greater than $500, within $45 of the mean, and identifying what constitutes an unusual stock price based on z-scores. Additionally, the assignment involves computing quartiles, assessing the normality assumption by constructing a histogram, and using the empirical rule to find statistically unusual price bounds.
Paper For Above instruction
The analysis of stock prices through the lens of normal distribution assumptions offers valuable insights into the behavior of financial data. In this assignment, we examine the closing prices of Google stock over a specific one-year period, apply statistical tools to interpret this data, and evaluate the normality assumption's validity. Utilizing Excel for calculations and data analysis, we will explore probabilities associated with stock prices relative to the mean, and identify prices that are statistically unusual.
First, the dataset comprises daily closing prices collected between November 20, 2012, and November 19, 2013. The primary objective is to analyze whether the stock prices follow a normal distribution pattern. Through Excel, plotting a histogram of the closing prices will provide a visual assessment of the distribution shape. The histogram's bell-shaped curve, along with the similarity of mean and median, and the approximate equal lengths of the interquartile ranges (Q1 to Q2 and Q2 to Q3), will support or challenge the normality assumption.
Next, we need to determine the mean and standard deviation of the dataset, which are essential for probability calculations. These statistical measures can be obtained efficiently using Excel functions such as =AVERAGE() and =STDEV.S(). Once we have these values, the first probability to analyze is: given the recent stock data, what is the probability that a randomly selected closing price was less than the mean? This can be answered intuitively utilizing the Empirical Rule, since approximately 50% of the data in a normal distribution lies below the mean, reinforced by the symmetry of the bell curve. This probability equals 0.5, assuming perfect normality. This simplified approach, based on the property that the normal
curve is symmetric about its mean, avoids explicit calculation of z-scores for this question.
The second probability involves the stock closing above $500. To compute this, calculate the mean and standard deviation of the dataset in Excel. Then, find the z-score corresponding to a closing price of $500: z = (x - mean) / standard deviation. Once the z-score (denoted as z1) is found, the probability that the stock closed at more than $500 is given by P(z > z1), which equals 1 - P(z < z1). Using Excel's NORM.S.DIST(z, TRUE) function, we obtain P(z < z1), and hence, the probability of exceeding $500.
The third task is to find the probability that the closing price was within $45 of the mean. This requires calculating two z-scores: one for (mean - 45) and another for (mean + 45). Using Excel, find these z-scores and their corresponding cumulative probabilities P(z < z1) and P(z < z2). The probability that the closing price falls within $45 of the mean is then the difference between these two probabilities: P = P(z < z2)P(z < z1). This calculation captures the proportion of data within a symmetric interval around the mean.
Regarding the assessment of unusual stock prices, a claim of buying at $700 per share can be evaluated by calculating its z-score: z = (700 - mean) / standard deviation. Empirical Rule states that points more than 2 standard deviations from the mean (z > 2 or z < -2) are considered unusual. If the z-score for $700 exceeds 2 in absolute value, the price is unusual. Since $700 is significantly higher than the average, the z-score’s magnitude will determine whether such a price is rare or typical.
For defining the statistical bounds of unusual prices, calculate the lower and upper bounds of normal variability using the empirical rule. Specifically, the normal distribution's typical range is within ±2 standard deviations from the mean. Therefore, the lower bound = mean - 2 × standard deviation, and the upper bound = mean + 2 × standard deviation. Prices outside this interval are statistically unusual, indicating a potential outlier in the data set.
Quartiles Q1, Q2, and Q3 provide further descriptive statistics of the data. Q1 (25th percentile), Q2 (50th percentile or median), and Q3 (75th percentile) offer insights into the data's spread and symmetry. These quartiles can be calculated using Excel functions: =QUARTILE(array, 1), =QUARTILE(array, 2), and =QUARTILE(array, 3). Comparing these quartile values helps in assessing skewness. If Q1 - Q2 and Q2Q3 are approximately equal, the data is more likely symmetric and consistent with normality; significant differences may suggest skewness or deviations from normality.
Finally, to evaluate whether the assumption of normality is valid, constructing a histogram of the dataset is essential. A bell-shaped, symmetric histogram supports normal distribution; however, biases, skewness, or
outliers may indicate deviations. Comparing the median with the mean further informs this assessment: if they are close, the data might approximate a normal distribution. Discrepancies between these measures and heterogeneous quartile spreads (differences between Q1, Q2, Q3) can indicate a non-normal distribution.
In conclusion, this statistical analysis demonstrates how normal distribution properties and Excel tools can be effectively employed to understand stock price behavior. While the Empirical Rule simplifies probability estimations, actual data visualization and calculations are crucial to validate assumptions. Recognizing outliers and unusual prices enables better financial decision-making. Such analysis illustrates the importance of statistical literacy in financial contexts and emphasizes the necessity for critical evaluation of distributional assumptions before applying probabilistic models.
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