Statistical Analysis with SAS @AccAnalyticsCPA ​

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I. Overview
This briefing covers fundamental business statistics concepts and an example of how statistical procedures (specifically, t-tests) are applied to analyze industry data. The textbook provides the theoretical background, definitions, and formulas for various statistical methods. The statistical output demonstrates the application of t-tests to compare means of variables for different industry groups.
II. Key Themes and Concepts from the Textbook
Descriptive Statistics: The textbook dedicates a large portion to descriptive statistics, which involves summarizing and presenting data. Key concepts include:
Measures of Central Tendency: Mean, median, and mode are discussed as ways to find the "center" of a dataset. "Formally the arithmetic mean is known as the first moment of the distribution. The second moment we will see is the variance, and skewness is the third moment."
Measures of Spread: Range, variance, and standard deviation quantify the variability within a dataset. "The Standard Deviation allows us to compare individual data or classes to the data set mean numerically."
Skewness: This measures the asymmetry of a distribution. "Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right."
Coefficient of Variation: Used to compare the degree of variation from one data series to another, even if the means are different. "CV = s / x̄ * 100 conditioned upon x̄ ≠ 0"
Probability: Introduces basic probability concepts like terminology, independent and mutually exclusive events, basic rules, contingency tables, and Venn Diagrams.
Discrete and Continuous Random Variables: Discusses different types of probability distributions, including: Binomial Distribution: Models the probability of successes in a fixed number of independent trials. "The n trials are independent and are repeated using identical conditions." "P(X = x) = ⎛ ⎝ n x ⎞ ⎠ pxqn − x"
Geometric Distribution: Models the number of trials needed to achieve the first success. "The trials are repeated until the first success." Poisson Distribution: Models the number of events occurring within a fixed interval of time or space. "P(x) = µ x e-µ / x !" Uniform Distribution: Where all values within a range are equally likely. "All values x are equally likely." "The mean is µ = a + b 2 and the standard deviation is σ = (b − a)2 / 12 ."
Exponential Distribution: Models the time until an event occurs. "pdf: f(x) = me(–mx) where x ≥ 0 and m > 0" Normal Distribution: A fundamental continuous probability distribution, often used to model real-world phenomena. "The normal probability density function, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped."
Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1. "Z ~ N(0, 1)" Z-score: A measure of how many standard deviations a data point is from the mean. "z = x – µ / σ or z = |x – µ| / σ" Central Limit Theorem: This crucial theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. "If the size n of the sample is sufficiently large, then X̄ ~ N⎛ ⎝µ, σ / n ⎞ ⎠ . If the size n of the sample is sufficiently large, then the distribution of the sample means will approximate a normal distribution regardless of the shape of the population."
Confidence Intervals: Provides a range of values likely to contain a population parameter with a certain level of confidence. "The confidence interval estimate has the format ( x- – EBM, x- + EBM) or the formula:" "X - − Zα ⎛ ⎝ σ / n ⎞ ⎠ ≤ µ ≤ X -+ Zα ⎛ ⎝ σ / n ⎞ ⎠"
Hypothesis Testing: A method for making decisions about population parameters based on sample data. Key elements include: Null and Alternative Hypotheses: Statements about the population parameter being tested. "H0: μ = 34; Ha: μ ≠ 34"
Type I and Type II Errors: Potential errors in hypothesis testing.
Test Statistic: A value calculated from sample data to determine whether to reject the null hypothesis.
P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. Chi-Square Distribution: Used for tests involving categorical data.
Goodness-of-Fit Test: Tests whether a sample distribution matches a population distribution.
Test of Independence: Tests whether two categorical variables are independent.
Test for Homogeneity: Tests whether different populations have the same distribution of a categorical variable. "χc 2 = Σ (O − E)2 / E where O = observed values and E = expected values"
F-Distribution and ANOVA (Analysis of Variance): Used to compare the means of two or more groups. "SSbetween = ∑ ⎡ ⎣ ⎢(s j) 2 / n j ⎤ ⎦ ⎥ − ⎛ ⎝∑ s j ⎞ ⎠ 2 / n" "F = MSbetween / MSwithin" Linear Regression and Correlation: Examines the linear relationship between two variables.
Correlation Coefficient (r): Measures the strength and direction of the linear relationship. "r = 1 / n − 1 Σ ⎛ ⎝X1i − X – 1 ⎞ ⎠ ⎛ ⎝X2i − X – 2 ⎞ ⎠ / sx1 sx2"
Regression Equation: Models the relationship between the independent and dependent variables.
Coefficient of Determination (R-squared): Represents the proportion of variance in the dependent variable explained by the independent variable.
Elasticity: Measures the responsiveness of one variable to changes in another. "ηp = ⎛ ⎝%∆Q⎞ ⎠ / (%∆P) = dQ / dP ⎛ ⎝ / P / Q ⎞ ⎠ = b⎛ / ⎝ P / Q ⎞ ⎠" III. Statistical Output Analysis ("Topic 2 example 2-results.pdf")
T-Tests: The document presents the results of t-tests comparing means of various variables between two industry groups (industry 53 and industry 54). The variables include: gvkey (Company identifier)
TA (Total Assets)
COGS (Cost of Goods Sold)
INVENTORY
TL (Total Liabilities)
SALES
SIC (Standard Industrial Classification code)
leverage (Financial leverage)
margin (Profit margin)
ind (Industry indicator)
Key Observations:
The output shows the means, medians, and sample sizes (N) for each variable within each industry. The TTEST Procedure output shows the results of the t-tests, including the means, standard deviations, standard errors, minimum and maximum values for TA and leverage in each industry. It also shows the difference in means (Diff (1-2)) and associated statistics. There are kernel density plots and Q-Q plots visualizing the distribution of TA, leverage, and margin for each industry. Example Interpretation: The T-test for Total Assets (TA) shows the following: Industry 53 (N=16): Mean TA = 27633.8, Std Dev = 57493.0
Industry 54 (N=22): Mean TA = 12795.4, Std Dev = 15710.4
Diff (1-2) = 14838.4 (This is the difference in means)
Pooled Std Error = 12815.0
These values, along with the degrees of freedom (not shown here, but would be in a complete output), would be used to calculate the t-statistic and p-value to determine if the difference in mean TA is statistically significant.
IV. Connections between the Sources
The textbook provides the foundational knowledge needed to understand and interpret the statistical output. For example, the textbook explains:
The purpose of a t-test: Testing for differences in means.
The meaning of key statistics: Mean, standard deviation, p-value, etc.
The concept of statistical significance: How to determine if an observed difference is likely due to chance or a real effect.
The use of visual aids like histograms and Q-Q plots: To assess the distribution of data.
V. Implications
By applying statistical techniques to real-world data, businesses can gain insights into their operations, markets, and competitors. Understanding statistical concepts is crucial for making informed decisions based on data analysis. T-tests can help compare different groups or segments, identify areas of strength and weakness, and track performance over time. Regression analysis can help to understand the relationship between different variables, such as sales and advertising, or costs and production volume. Hypothesis testing can be used to test assumptions and make predictions about the future.
VI. Next Steps
To fully analyze the statistical output, the t-statistics and p-values need to be examined to determine statistical significance. Further investigation into the specific industries represented by SIC codes 53 and 54 would provide context for the observed differences. Consider performing other statistical tests (e.g., regression analysis, ANOVA) to explore relationships between variables in more detail. Use the business statistics textbook to learn more about the statistical tests used in the output.