Business Quantitative Methods: Statistics

• Business quantitative methods involve using mathematical and statistical techniques to analyze data and solve business problems.
• Statistical analysis is a key component, focusing on collecting, analyzing, interpreting, and presenting data to support decision-making in various business functions.

Descriptive Statistics

• Descriptive statistics summarize and present data in a meaningful way.
• Measures of central tendency include the mean (average), median (middle value), and mode (most frequent value).
• The mean is sensitive to outliers, while the median is more robust.
• Measures of dispersion include range, variance, and standard deviation.
• Range is the difference between the maximum and minimum values.
• Variance measures the average squared deviation from the mean.
• Standard deviation is the square root of the variance, providing a more interpretable measure of spread.
• Skewness measures the asymmetry of a distribution.
• Positive skewness indicates a longer tail on the right.
• Negative skewness indicates a longer tail on the left.
• Kurtosis measures the "tailedness" of a distribution.
• High kurtosis indicates heavy tails and more outliers.
• Low kurtosis indicates light tails and fewer outliers.

Probability

• Probability is the measure of the likelihood that an event will occur.
• Basic probability rules:
• The probability of any event is between 0 and 1.
• The sum of the probabilities of all possible outcomes is 1.
• Conditional probability is the probability of an event occurring given that another event has already occurred.
• Denoted as P(A|B), the probability of A given B.
• Independence: Two events are independent if the occurrence of one does not affect the probability of the other.
• P(A|B) = P(A)
• Bayes' Theorem updates the probability of an event based on new evidence.
• P(A|B) = [P(B|A) * P(A)] / P(B)

Probability Distributions

• Discrete probability distributions deal with countable outcomes.
• The binomial distribution models the probability of a certain number of successes in a fixed number of trials.
• Requires independent trials, each with a constant probability of success.
• The Poisson distribution models the number of events occurring in a fixed interval of time or space.
• Assumes events occur independently and at a constant average rate.
• Continuous probability distributions deal with outcomes that can take on any value within a range.
• The normal distribution is a symmetric, bell-shaped distribution.
• Characterized by its mean and standard deviation.
• Many natural phenomena follow a normal distribution.
• The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
• Used to calculate probabilities for any normal distribution by converting values to z-scores.
• The exponential distribution models the time until an event occurs.
• Often used in reliability analysis and queuing theory.

Sampling and Estimation

• Sampling involves selecting a subset of a population to gather data.
• Random sampling ensures that each member of the population has an equal chance of being selected.
• Stratified sampling divides the population into subgroups (strata) and takes a random sample from each stratum.
• Cluster sampling divides the population into clusters and randomly selects some clusters to sample.
• Sampling error is the difference between a sample statistic and the corresponding population parameter.
• A point estimate is a single value used to estimate a population parameter.
• The sample mean is a point estimate of the population mean.
• An interval estimate is a range of values used to estimate a population parameter.
• A confidence interval provides a range within which the population parameter is likely to lie, with a specified level of confidence.
• The margin of error determines the width of the confidence interval.
• Factors affecting the margin of error include the sample size, the standard deviation, and the level of confidence.
• A larger sample size reduces the margin of error.
• A higher standard deviation increases the margin of error.
• A higher level of confidence increases the margin of error.

Hypothesis Testing

• Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis.
• The null hypothesis (H0) is a statement about a population parameter that is assumed to be true.
• The alternative hypothesis (H1) is a statement that contradicts the null hypothesis.
• A test statistic is calculated from the sample data to assess the evidence against the null hypothesis.
• A p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
• If the p-value is less than the significance level (alpha), the null hypothesis is rejected.
• A Type I error occurs when the null hypothesis is rejected when it is actually true (false positive).
• A Type II error occurs when the null hypothesis is not rejected when it is actually false (false negative).
• The power of a test is the probability of correctly rejecting the null hypothesis when it is false (1 - probability of Type II error).
• Common hypothesis tests include:
• t-tests: Used to compare means of one or two groups.
• z-tests: Used to compare means when the population standard deviation is known.
• ANOVA (Analysis of Variance): Used to compare means of three or more groups.
• Chi-square tests: Used to test for association between categorical variables.

Regression Analysis

• Regression analysis examines the relationship between a dependent variable and one or more independent variables.
• Simple linear regression involves one independent variable.
• The equation of the regression line is y = a + bx, where y is the dependent variable, x is the independent variable, a is the intercept, and b is the slope.
• Multiple linear regression involves two or more independent variables.
• The equation is y = a + b1x1 + b2x2 + ... + bnxn.
• The coefficient of determination (R-squared) measures the proportion of variance in the dependent variable that is explained by the independent variables.
• Ranges from 0 to 1, with higher values indicating a better fit.
• Residuals are the differences between the observed values and the predicted values.
• Assumptions of linear regression:
• Linearity: The relationship between the variables is linear.
• Independence: The residuals are independent.
• Homoscedasticity: The variance of the residuals is constant across all levels of the independent variables.
• Normality: The residuals are normally distributed.
• Violation of these assumptions can lead to biased or inefficient estimates.
• Dummy variables are used to include categorical variables in regression models.

Time Series Analysis

• Time series analysis involves analyzing data points collected over time.
• Components of a time series:
• Trend: The long-term movement in the data.
• Seasonality: Regular, repeating patterns that occur within a year.
• Cyclical variations: Longer-term patterns that are not necessarily periodic.
• Irregular variations: Random fluctuations.
• Moving averages smooth out short-term fluctuations to reveal the underlying trend.
• Exponential smoothing is a method for forecasting future values based on past values, with more weight given to recent data.
• Autocorrelation measures the correlation between a time series and its lagged values.
• Stationarity: A time series is stationary if its statistical properties (mean, variance, autocorrelation) do not change over time.
• Many time series models require stationarity.
• ARIMA (Autoregressive Integrated Moving Average) models are used to forecast time series data.
• Involve autoregressive (AR), integrated (I), and moving average (MA) components.

Decision Analysis

• Decision analysis is a framework for making decisions under uncertainty.
• Decision alternatives are the different courses of action available to the decision-maker.
• States of nature are the possible events that could occur, which are outside the control of the decision-maker.
• Payoffs are the outcomes resulting from each combination of decision alternative and state of nature.
• Decision criteria:
• Expected Monetary Value (EMV): The weighted average of the payoffs for each alternative, using the probabilities of the states of nature as weights.
• Expected Opportunity Loss (EOL): The expected value of the opportunity loss for each alternative.
• Maximax: Choose the alternative with the maximum possible payoff.
• Maximin: Choose the alternative with the maximum of the minimum payoffs.
• Decision trees are graphical representations of the decision-making process.
• Sensitivity analysis examines how the optimal decision changes when the probabilities or payoffs are changed.
• Value of Information:
• Expected Value of Perfect Information (EVPI): The maximum amount a decision-maker would be willing to pay for perfect information about the future.
• Calculated as the difference between the expected value with perfect information and the expected value without perfect information.