Math Formulas for Business Statistics

Binomial Distribution

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

• Notation:

• Mean:

• Standard Deviation: , where

• Example: If a factory produces light bulbs with a 5% defect rate (), and 100 bulbs are tested (), the expected number of defective bulbs is .

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a large sample size, the sampling distribution of the sample mean or sample proportion will be approximately normal, regardless of the population's distribution.

• Sample Mean:

• Sample Proportion:

• Example: If the population mean is 50 and standard deviation is 10, for , the sampling distribution of the mean is .

Confidence Intervals

A confidence interval estimates a population parameter with a specified level of confidence (e.g., 95%).

• For Mean (Known Standard Deviation):

• For Mean (Unknown Standard Deviation):

• For Proportion:

• Example: For a sample mean of 100, , , and 95% confidence (): .

Hypothesis Testing Test Statistics

Hypothesis testing involves comparing a sample statistic to a hypothesized population value using a test statistic.

• z-test for Mean:

• t-test for Mean:

• z-test for Proportion:

• Example: Testing if the average sales is different from using sample data.

Linear Regression

Linear regression models the relationship between a dependent variable and one or more independent variables.

• Confidence Interval for Mean Response at : , where

• Prediction Interval for Response at :

• Example: Predicting sales () based on advertising spend ().

Excel Formulas for Business Statistics

Basic Operations

• Addition: +

• Subtraction: -

• Multiplication: *

• Division: /

• Square Root: sqrt()

Descriptive Statistics

• Maximum: =max()

• Minimum: =min()

• Sample Mean: =average()

• Sample Standard Deviation: =stdev()

• Sample Median: =median()

• Histogram: Data analysis toolpak → Histogram

Probability Distributions

• Binomial Distribution:

  • P(x ≤ a): =BINOM.DIST(a, n, p, TRUE)

  • P(x = a): =BINOM.DIST(a, n, p, FALSE)

• Normal Distribution:

  • P(x ≤ a): =norm.dist(a, µ, σ, true)

• Standard Normal:

  • zα: =norm.s.inv(1-α)

• tα: =t.inv(1-α, df)

Regression and Correlation

• Fit Linear Regression Model: Data analysis toolpak → Regression

• Correlation Coefficient (r): =correl(x data, y data)

• SSxx (Sum of Squares): =devsq(x data)

Summary Table: Key Formulas and Excel Functions

Additional info: The formulas and Excel functions listed are foundational for business statistics, covering descriptive statistics, probability distributions, hypothesis testing, and regression analysis. Students should be familiar with both the mathematical concepts and their implementation in Excel for practical data analysis.