Business Statistics (QUAN 2600 | Weber State)

[[#Chapter 1: Introduction to Statistics]]
[[#Chapter 2: Data Visualization]]
[[#Chapter 3: Numerical Measurements]]
[[#Chapter 4: Probability]]
[[#Chapter 5: Discrete Probability Distributions]]
[[#Chapter 6: Continuous Probability Distributions]]
[[#Chapter 7: Sampling and Sampling Distributions]]
[[#Chapter 8: Interval Estimation]]
[[#Chapter 9: Hypothesis Testing]]
[[#Key Formulas Summary]]

Chapter 1: Introduction to Statistics

Core Concepts

Statistics = Organizing disorganized data to understand and communicate information

Types of Statistics:

Descriptive Statistics: Summary of data (tabular, graphical, numerical)
Statistical Inference: Using sample data to make estimates about populations

Data and Variables

Data Sources:

Experimental Data: Randomly assigned control/treatment groups (causal relationships)
Observational Data: Non-experimental observations (surveys, studies)
Existing Data: Internal records, government data, public databases

Key Terms:

Element: Entity on which data are collected (IDs)
Variable: Characteristic of interest (columns)
Observation: Complete set of variables for an entity (rows)
Population: All elements of interest
Sample: Subset of population
Census: Data collection for entire population

Scales of Measurement

Categorical (Qualitative):

Nominal: Labels with no order (names, colors)
Ordinal: Labels with meaningful order (rankings, grades)

Quantitative (Numerical):

Interval: Numbers with fixed units, no true zero (temperature in Celsius)
Ratio: Numbers with true zero (height, weight, income)

Analytics Types

Descriptive: What happened in the past
Predictive: Using models to forecast future
Prescriptive: Optimal course of action

Chapter 2: Data Visualization

Frequency Distributions

Basic Concepts:

Frequency: Number of observations in each category
Relative Frequency: Frequency ÷ Total observations
Percent Frequency: Relative frequency × 100

For Quantitative Data:

Use 5-20 classes
Class Width = Range ÷ Number of classes (round up)
Cumulative Frequency: ≤ upper limit of class

Charts and Graphs

Categorical Data:

Bar Charts: Fixed width bars, can be sorted
Side-by-Side Bar Charts: Compare two variables
Stacked Bar Charts: Variables stacked in bars
Pie Charts: 50% = 180°, 1% = 3.6°

Quantitative Data:

Dot Plots: Simple summary for small datasets
Histograms: Connected rectangles showing distribution
Stem-and-Leaf: Shows rank order and shape simultaneously

Two Variables:

Crosstabulations: Tabular summary of two variables
Scatter Diagrams: Relationship between quantitative variables
Trendlines: Approximate relationship line

Distribution Shapes

Symmetric: Bell-shaped, mean = median
Positive Skew: Tail extends right, mean > median
Negative Skew: Tail extends left, mean < median

Chapter 3: Numerical Measurements

Measures of Location

Mean

\bar{x} = \frac{\sum x_{i}}{n}

Weighted Mean

{\bar{x}}_{w} = \frac{\sum x_{i} w_{i}}{\sum w_{i}}

Geometric Mean

\sqrt[n]{\prod_{i = 1}^{n} x_{i}}

Used for growth rates
Growth factor = 1 + return percentage

Median

Middle value when data is ordered
If n is even: average of two middle values

Mode

Most frequently occurring value
Can be bimodal or multimodal

Percentiles

L_{p} = \frac{p}{100} (n + 1)

Quartiles (Q): 25% increments
Deciles (D): 10% increments
Five-Number Summary: Min, Q₁, Q₂, Q₃, Max

Z-Score (Standardized Value)

z = \frac{x_{i} - \bar{x}}{s}

Measures standard deviations from mean
|z| > 3 indicates outlier

Measures of Variability

Range

Range = Largest - Smallest

Interquartile Range (IQR)

IQR = Q_{3} - Q_{1}

Variance

Population Variance: $$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$$

Sample Variance: $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$$

Standard Deviation

σ = \sqrt{σ^{2}} or s = \sqrt{s^{2}}

Coefficient of Variation

C V = \frac{Standard Deviation}{Mean} \times 100

Outlier Detection Methods

Z-Score Method: |z| > 3 IQR Method:

Lower limit = Q₁ - 1.5(IQR)
Upper limit = Q₃ + 1.5(IQR)

Distribution Rules

Chebyshev's Theorem

At least 1 - \frac{1}{z^{2}} of data within z standard deviations

Applies to any distribution
z must be > 1

Empirical Rule (Bell-Shaped Distributions)

68.26% within 1σ
95.44% within 2σ
99.74% within 3σ

Measures of Association

Covariance

Cov (X, Y) = \frac{\sum (x_{i} - \bar{x}) (y_{i} - \bar{y})}{n - 1}

Correlation Coefficient

r = \frac{Cov (X, Y)}{σ_{X} σ_{Y}}

Range: -1 ≤ r ≤ 1
+1: Perfect positive linear relationship
-1: Perfect negative linear relationship
0: No linear relationship

Chapter 4: Probability

Basic Concepts

Probability Scale: 0 ≤ P(E) ≤ 1

0 = impossible
0.5 = equally likely
1 = certain

Sample Space: All possible outcomes Sample Point: Single outcome Event: Collection of sample points

Counting Rules

Multi-part Experiments: $$\text{Total Outcomes} = (n_1)(n_2)...(n_k)$$

Combinations (order doesn't matter): $$C_n^r = \frac{n!}{r!(n-r)!}$$

Permutations (order matters): $$P_n^r = \frac{n!}{(n-r)!}$$

With Replacement: $$x^y$$

Assigning Probabilities

Methods:

Classical: Equal probability for all outcomes
Relative Frequency: Based on historical data
Subjective: Based on belief/judgment

Probability Relationships

Complement

P (A^{c}) = 1 - P (A)

Addition Law

P (A \cup B) = P (A) + P (B) - P (A \cap B)

For Mutually Exclusive Events: $$P(A \cup B) = P(A) + P(B)$$

Conditional Probability

P (A | B) = \frac{P (A \cap B)}{P (B)}

Multiplication Law

P (A \cap B) = P (A) \cdot P (B | A)

For Independent Events: $$P(A \cap B) = P(A) \cdot P(B)$$

Chapter 5: Discrete Probability Distributions

Random Variables

Discrete Random Variable: Countable outcomes

Can be finite or infinite

Probability Distribution f(x):

f(x) ≥ 0 for all x
Σf(x) = 1

Expected Value and Variance

Expected Value (Mean)

E (X) = μ = \sum x \cdot f (x)

Variance

σ^{2} = \sum (x_{i} - μ)^{2} f (x_{i})

Standard Deviation

σ = \sqrt{σ^{2}}

Bivariate Distributions

Linear Combination

E (a X + b Y) = a E (X) + b E (Y)

Combined Variance

Var (a X + b Y) = a^{2} Var (X) + b^{2} Var (Y) + 2 a b \cdot Cov (X, Y)

Correlation Coefficient

ρ = \frac{σ_{X Y}}{σ_{X} σ_{Y}}

Binomial Distribution

Properties:

n identical trials
Two outcomes per trial (success/failure)
Constant probability p
Independent trials

Binomial Probability

f (x) = \frac{n!}{x! (n - x)!} p^{x} (1 - p)^{n - x}

Where:

x = number of successes
n = number of trials
p = probability of success

Binomial Expected Value

E (X) = n p

Binomial Variance

Var (X) = n p (1 - p)

Chapter 6: Continuous Probability Distributions

Continuous Distributions

Key Concepts:

Probability Density Function: Total area = 1
Probability = area under curve
P(X = specific value) = 0

Uniform Distribution

Probability Density Function

f (x) = \frac{1}{b - a} for a \leq x \leq b

Expected Value

E (X) = \frac{a + b}{2}

Variance

Var (X) = \frac{(b - a)^{2}}{12}

Normal Distribution

Probability Density Function

f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}

Properties:

μ = mean (also median and mode)
σ = standard deviation (controls width)
Symmetric, bell-shaped

Standard Normal Distribution

μ = 0, σ = 1
Denoted as Z

Converting to Standard Normal

Z = \frac{X - μ}{σ}

Empirical Rule

68.27% within 1σ
95.45% within 2σ
99.73% within 3σ

Chapter 7: Sampling and Sampling Distributions

Sampling Concepts

Key Terms:

Element: Entity on which data are collected
Population (N): All elements of interest
Sample (n): Subset of population
Frame: List of elements for sampling
Simple Random Sample: Each element has equal selection chance

Population Types:

Finite: Can count all elements
Infinite: Cannot count all elements (treat as infinite if n/N ≤ 0.05)

Point Estimation

Point Estimators:

$\bar{x}$ estimates μ (population mean)
s estimates σ (population standard deviation)
$\bar{p}$ estimates p (population proportion)

Sample Mean

\bar{x} = \frac{\sum x_{i}}{n}

Sample Standard Deviation

s = \sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n - 1}}

Sample Proportion

\bar{p} = \frac{x}{n}

Sampling Distribution of $\bar{x}$

Properties:

E( $\bar{x}$ ) = μ (unbiased estimator)
When population is normal: $\bar{x}$ is normal for any n
When population is not normal: $\bar{x}$ is approximately normal for large n (Central Limit Theorem)

Standard Error of Mean

Infinite Population: $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Finite Population: $$\sigma_{\bar{x}} = \sqrt{\frac{N-n}{N-1}} \cdot \frac{\sigma}{\sqrt{n}}$$

Central Limit Theorem

For large n (≥30), $\bar{x}$ is approximately normal
For highly skewed data, use n ≥ 50

Sampling Distribution of $\bar{p}$

Properties:

E( $\bar{p}$ ) = p
Approximately normal when np ≥ 5 and n(1-p) ≥ 5

Standard Error of Proportion

Infinite Population: $$\sigma_{\bar{p}} = \sqrt{\frac{p(1-p)}{n}}$$

Finite Population: $$\sigma_{\bar{p}} = \sqrt{\frac{N-n}{N-1}} \cdot \sqrt{\frac{p(1-p)}{n}}$$

Chapter 8: Interval Estimation

Confidence Intervals

General Form: Point Estimator ± Margin of Error

Confidence Level (1-α):

90%: z₀.₀₅ = 1.645
95%: z₀.₀₂₅ = 1.96
99%: z₀.₀₀₅ = 2.576

Population Mean (σ Known)

Confidence Interval

\bar{x} \pm z_{α / 2} \frac{σ}{\sqrt{n}}

Margin of Error

E = z_{α / 2} \frac{σ}{\sqrt{n}}

Population Mean (σ Unknown)

Confidence Interval

\bar{x} \pm t_{α / 2} \frac{s}{\sqrt{n}}

Use t-distribution with df = n-1

Sample Size Determination

For Mean

n = \frac{(z_{α / 2})^{2} σ^{2}}{E^{2}}

For Proportion

n = \frac{(z_{α / 2})^{2} p^{*} (1 - p^{*})}{E^{2}}

Where p* is planning value (use 0.5 if unknown for largest sample size)

Population Proportion

Confidence Interval

\bar{p} \pm z_{α / 2} \sqrt{\frac{\bar{p} (1 - \bar{p})}{n}}

Margin of Error

E = z_{α / 2} \sqrt{\frac{\bar{p} (1 - \bar{p})}{n}}

Requirements: n $\bar{p}$ ≥ 5 and n(1- $\bar{p}$ ) ≥ 5

Chapter 9: Hypothesis Testing

Hypothesis Structure

Null Hypothesis (H₀): Tentative assumption

Contains =, ≤, or ≥
Assumed true until evidence suggests otherwise

Alternative Hypothesis (Hₐ): Deviation from assumption

Contains ≠, <, or >
What we're trying to prove

Types of Errors

Type I Error (α):

Rejecting H₀ when it's true
Level of significance
Can be controlled

Type II Error (β):

Failing to reject H₀ when it's false
Difficult to control

Hypothesis Testing Steps

State Hypotheses (H₀ and Hₐ)
Choose Significance Level (α)
Calculate Test Statistic
Find P-value
Make Decision (Compare p-value to α)

Decision Rule:

If p-value < α: Reject H₀
If p-value ≥ α: Do not reject H₀

Test Statistics

Population Mean (σ Known)

z = \frac{\bar{x} - μ_{0}}{σ / \sqrt{n}}

P-value Calculation

One-tailed test: P(Z > z) or P(Z < z) Two-tailed test: 2 × P(Z > |z|)

Test Types

One-tailed: H₀ contains ≤ or ≥, Hₐ contains < or >
Two-tailed: H₀ contains =, Hₐ contains ≠

Key Formulas Summary

Descriptive Statistics

Measure	Formula
Sample Mean	$\bar{x} = \frac{\sum x_{i}}{n}$
Sample Variance	$s^{2} = \frac{\sum (x_{i} - \bar{x})^{2}}{n - 1}$
Sample Standard Deviation	$s = \sqrt{s^{2}}$
Z-Score	$z = \frac{x_{i} - \bar{x}}{s}$
Correlation	$r = \frac{Cov (X, Y)}{σ_{X} σ_{Y}}$

Probability

Concept	Formula
Combinations	$C_{n}^{r} = \frac{n!}{r! (n - r)!}$
Permutations	$P_{n}^{r} = \frac{n!}{(n - r)!}$
Addition Law	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Conditional	$P(A
Binomial	$f (x) = \frac{n!}{x! (n - x)!} p^{x} (1 - p)^{n - x}$

Sampling Distributions

Distribution	Standard Error
Mean (Infinite)	$σ_{\bar{x}} = \frac{σ}{\sqrt{n}}$
Mean (Finite)	$σ_{\bar{x}} = \sqrt{\frac{N - n}{N - 1}} \cdot \frac{σ}{\sqrt{n}}$
Proportion (Infinite)	$σ_{\bar{p}} = \sqrt{\frac{p (1 - p)}{n}}$
Proportion (Finite)	$σ_{\bar{p}} = \sqrt{\frac{N - n}{N - 1}} \cdot \sqrt{\frac{p (1 - p)}{n}}$

Confidence Intervals

Parameter	Confidence Interval
Mean (σ known)	$\bar{x} \pm z_{α / 2} \frac{σ}{\sqrt{n}}$
Mean (σ unknown)	$\bar{x} \pm t_{α / 2} \frac{s}{\sqrt{n}}$
Proportion	$\bar{p} \pm z_{α / 2} \sqrt{\frac{\bar{p} (1 - \bar{p})}{n}}$

Sample Size

Parameter	Sample Size Formula
Mean	$n = \frac{(z_{α / 2})^{2} σ^{2}}{E^{2}}$
Proportion	$n = \frac{(z_{α / 2})^{2} p^{} (1 - p^{})}{E^{2}}$

Hypothesis Testing

Test	Test Statistic
Mean (σ known)	$z = \frac{\bar{x} - μ_{0}}{σ / \sqrt{n}}$
Mean (σ unknown)	$t = \frac{\bar{x} - μ_{0}}{s / \sqrt{n}}$

Common Z-Values

90% Confidence: z₀.₀₅ = 1.645
95% Confidence: z₀.₀₂₅ = 1.96
99% Confidence: z₀.₀₀₅ = 2.576

Important Notes

When to Use Z vs T

Use Z when: σ is known, or n ≥ 30 with s
Use T when: σ is unknown and n < 30
Degrees of freedom: df = n - 1

Normal Distribution Conditions

Sampling distribution of $\bar{x}$ : Normal population OR n ≥ 30 (CLT)
Sampling distribution of $\bar{p}$ : np ≥ 5 AND n(1-p) ≥ 5
Finite population correction: Use when n/N > 0.05

Key Concepts to Remember

Unbiased estimator: E(estimator) = parameter
Central Limit Theorem: n ≥ 30 for normal approximation
Type I error: α = P(reject H₀ | H₀ true)
P-value: Probability of observing test statistic or more extreme
Confidence level: 1 - α

Business Statistics (QUAN 2600 | Weber State)

Table of Contents

Chapter 1: Introduction to Statistics

Core Concepts

Data and Variables

Scales of Measurement

Analytics Types

Chapter 2: Data Visualization

Frequency Distributions

Charts and Graphs

Distribution Shapes

Chapter 3: Numerical Measurements

Measures of Location

Mean

Weighted Mean

Geometric Mean

Median

Mode

Percentiles

Z-Score (Standardized Value)

Measures of Variability

Range

Interquartile Range (IQR)

Variance

Standard Deviation

Coefficient of Variation

Outlier Detection Methods

Distribution Rules

Chebyshev's Theorem

Empirical Rule (Bell-Shaped Distributions)

Measures of Association

Covariance

Correlation Coefficient

Chapter 4: Probability

Basic Concepts

Counting Rules

Assigning Probabilities

Probability Relationships

Complement

Addition Law

Conditional Probability

Multiplication Law

Chapter 5: Discrete Probability Distributions

Random Variables

Expected Value and Variance

Expected Value (Mean)

Variance

Standard Deviation

Bivariate Distributions

Linear Combination

Combined Variance

Correlation Coefficient

Binomial Distribution

Binomial Probability

Binomial Expected Value

Binomial Variance

Chapter 6: Continuous Probability Distributions

Continuous Distributions

Uniform Distribution

Probability Density Function

Expected Value

Variance

Normal Distribution

Probability Density Function

Standard Normal Distribution

Converting to Standard Normal

Empirical Rule

Chapter 7: Sampling and Sampling Distributions

Sampling Concepts

Point Estimation

Sample Mean

Sample Standard Deviation

Sample Proportion

Sampling Distribution of x¯

Standard Error of Mean

Central Limit Theorem

Sampling Distribution of p¯

Standard Error of Proportion

Chapter 8: Interval Estimation

Confidence Intervals

Sampling Distribution of $\bar{x}$

Sampling Distribution of $\bar{p}$