Business Statistics Study Guide: Chapters 1–4 (MAT 137)

Provided Formulas

Key Statistical Formulas

Understanding and applying statistical formulas is essential for analyzing data in business statistics. Below are the main formulas you need to know for Test 1:

• Sample Variance: Measures the average squared deviation from the mean in a sample. $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}$

• Expected Value (Discrete Random Variable): The mean value expected from a probability distribution. $E(x) = \sum x p(x)$

• Variance (Discrete Random Variable): Measures the spread of a probability distribution. $\sigma^2 = E[(x - \mu)^2] = \sum (x - \mu)^2 p(x)$

Chapter 1: Statistics, Data, and Statistical Thinking

Descriptive vs. Inferential Statistics

Statistics is divided into two main branches: descriptive and inferential statistics.

• Descriptive Statistics: Uses numerical and graphical methods to summarize and present information from a data set.

• Inferential Statistics: Uses sample data to make estimates, predictions, or generalizations about a population.

Key Definitions

• Experimental Unit: The object or entity being studied.

• Population: The complete set of units under investigation.

• Variable: A characteristic or property measured or observed.

• Sample: A subset of the population selected for analysis.

Types of Data

• Quantitative Data: Numerical values (e.g., income, age).

• Qualitative Data: Non-numerical categories (e.g., political party, breed).

Chapter 2: Methods for Describing Sets of Data

Frequency Distributions and Histograms

Frequency distributions and histograms are used to organize and visualize data.

• Class: Categories for qualitative data or intervals for quantitative data.

• Class Frequency: Number of observations in a class.

• Class Relative Frequency: Class frequency divided by total observations.

• Class Percentage: Relative frequency multiplied by 100.

Measures of Central Tendency

• Mean: Arithmetic average of a data set. $\bar{x} = \frac{\sum x_i}{n}$

• Median: Middle value when data is ordered.

Skewness and Symmetry

• Skewed Right: Mean > Median

• Skewed Left: Mean < Median

• Symmetrical: Mean = Median

Range, Variance, and Standard Deviation

• Range: Difference between largest and smallest values.

• Sample Variance: Measures variability in a sample.

• Sample Standard Deviation: Positive square root of sample variance. $s = \sqrt{s^2}$

The larger the variance or standard deviation, the more variable the data are. Sample statistics (s, s²) are used to estimate population parameters (σ, σ²).

Empirical Rule

• 68% of data falls within 1 standard deviation of the mean.

• 95% within 2 standard deviations.

• 99.7% within 3 standard deviations.

Z-Scores

Z-scores measure how many standard deviations a value is from the mean.

• Sample z-score: $z = \frac{x - \bar{x}}{s}$

• Population z-score: $z = \frac{x - \mu}{\sigma}$

Percentiles

• The pth percentile is the value below which p% of the data falls.

• Example: 90th percentile means 90% of data is below that value.

Chapter 3: Probability

Basic Probability Concepts

Probability quantifies the likelihood of events occurring in an experiment.

• Probability: A number between 0 and 1 representing the chance of an event.

• Experiment: A process that leads to a single, unpredictable outcome.

• Sample Point: The most basic outcome of an experiment.

• Sample Space: The set of all possible sample points.

• Event: A specific collection of sample points.

Probability Rules

• All sample point probabilities must be between 0 and 1.

• The sum of all sample point probabilities in a sample space must equal 1.

Finding Probabilities

• Probability of an event is its relative frequency in repeated experiments.

• Example: Flipping a fair coin, probability of heads = 0.5.

Set Operations and Complements

• The probability of complementary events sums to 1.

Chapter 4: Random Variables and Probability Distributions

Random Variables

A random variable assigns numerical values to outcomes of an experiment.

• Discrete Random Variable: Takes countable values.

• Continuous Random Variable: Takes values in intervals (uncountable).

Probability Distribution Requirements

• $p(x) \geq 0$ for all values of x

• $\sum p(x) = 1$ (sum over all possible values of x)

Expected Value and Variance

• Expected Value (Mean): $E(x) = \sum x p(x)$

• Variance: $\sigma^2 = \sum (x - \mu)^2 p(x)$

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Characteristics:

  • n identical trials

  • Two possible outcomes per trial: success (S) or failure (F)

  • Probability of success (p) remains constant

  • Trials are independent

• Probability Distribution Formula: $p(x) = \binom{n}{x} p^x q^{n-x}$ where $q = 1 - p$, $n$ = number of trials, $x$ = number of successes, $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Mean, Variance, and Standard Deviation of Binomial Distribution

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard Deviation: $\sigma = \sqrt{npq}$

Excel Functions for Binomial Distribution

BINOM.DIST Function

• BINOM.DIST(a, n, p, TRUE): Computes cumulative probability of up to a successes in n trials.

• BINOM.DIST(a, n, p, FALSE): Computes probability of exactly a successes in n trials.

• Arguments:

  • a: Number of successes

  • n: Number of trials

  • p: Probability of success