MAT 137 Midterm Study Guide: Chapters 1–5 (Business Statistics)

Statistics, Data, and Statistical Thinking

Introduction to Statistics

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It is fundamental for making informed decisions in business and research.

• Definition: Statistics involves methods for gathering data, summarizing information, and drawing conclusions.

• Applications: Used in business forecasting, quality control, market research, and more.

• Types of Statistics: Descriptive (summarizing data) and Inferential (drawing conclusions from data).

• Example: Calculating the average sales for a company over a year.

Collecting Data & Sampling Methods

Collecting data accurately is crucial for valid statistical analysis. Sampling methods determine how representative the data is.

• Population vs. Sample: A population is the entire group of interest; a sample is a subset used for analysis.

• Sampling Methods:

  • Random Sampling: Every member has an equal chance of selection.

  • Stratified Sampling: Population divided into subgroups, samples taken from each.

  • Cluster Sampling: Population divided into clusters, entire clusters are sampled.

  • Systematic Sampling: Every nth member is selected.

• Example: Surveying 100 customers chosen randomly from a database.

Methods for Describing Sets of Data

Visualizing Qualitative vs. Quantitative Data

Data can be classified as qualitative (categorical) or quantitative (numerical). Visualization helps in understanding data distribution.

• Qualitative Data: Describes categories or qualities (e.g., colors, brands).

• Quantitative Data: Measures quantities (e.g., sales figures, heights).

• Visualization Tools:

  • Bar Charts: For qualitative data.

  • Histograms: For quantitative data.

• Example: A histogram showing the distribution of employee ages.

Frequency Distributions & Histograms

Frequency distributions summarize data by showing how often each value occurs. Histograms graphically represent these distributions.

• Frequency Distribution: Table listing values and their frequencies.

• Histogram: Bar graph showing frequency of quantitative data intervals.

• Example: Frequency table of sales per week.

Measures of Central Tendency and Spread

Central tendency and spread describe the center and variability of data.

• Mean: The average value. $\text{Mean} = \frac{\sum x_i}{n}$

• Median: The middle value when data is ordered.

• Standard Deviation: Measures spread around the mean. $\text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• Interpreting Standard Deviation: Higher values indicate more variability.

• Percentiles & Quartiles: Indicate relative standing in data. Quartiles: Divide data into four equal parts.

• Example: Calculating the median income of a group.

Probability

Basic Concepts of Probability

Probability quantifies the likelihood of events occurring.

• Probability: Value between 0 and 1 representing chance of an event.

• Complement: Probability of event not occurring. $P(A') = 1 - P(A)$

• Addition Rule: For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$

• Contingency Tables: Used to analyze relationships between categorical variables.

• Example: Probability of drawing a red card from a deck.

Random Variables and Probability Distributions

Discrete Random Variables & Binomial Distribution

Random variables represent outcomes of random processes. Discrete random variables take specific values.

• Discrete Random Variable: Can only take certain values (e.g., number of sales).

• Binomial Distribution: Models number of successes in fixed trials. $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Finding Binomial Probabilities: Can use Excel functions like BINOM.DIST.

• Example: Probability of getting 3 heads in 5 coin tosses.

Normal Distribution

The normal distribution is a continuous, bell-shaped curve describing many natural phenomena.

• Standard Normal Distribution: Mean 0, standard deviation 1.

• Non-Standard Normal Distribution: Any mean and standard deviation.

• Finding Probabilities, Z Values, and X Values: Use Excel functions like NORM.DIST and NORM.INV.

• Formula: $Z = \frac{X - \mu}{\sigma}$

• Example: Calculating the probability that a randomly selected employee earns more than $50,000.

Sampling Distributions

Sampling Distribution of the Sample Mean & Central Limit Theorem

Sampling distributions describe the distribution of statistics (like the mean) from repeated samples. The Central Limit Theorem (CLT) is fundamental in inferential statistics.

• Sampling Distribution: Distribution of a statistic (e.g., sample mean) over many samples.

• Central Limit Theorem: For large samples, the sampling distribution of the mean is approximately normal, regardless of population shape.

• Formula: $\mu_{\bar{x}} = \mu$ and $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Distribution of Sample Mean: Can be calculated using Excel.

• Example: Estimating the average height of students from repeated samples.

Sampling Distribution of Sample Proportion

The sampling distribution of the sample proportion describes the variability of proportions from repeated samples.

• Sample Proportion: $\hat{p} = \frac{x}{n}$

• Mean and Standard Deviation: $\mu_{\hat{p}} = p$ and $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Example: Proportion of customers who prefer a new product.

Summary Table: Key Concepts and Tools

Topic	Definition	Example/Tool
Mean	Average value	Excel: AVERAGE()
Median	Middle value	Excel: MEDIAN()
Standard Deviation	Spread of data	Excel: STDEV()
Binomial Probability	Successes in trials	Excel: BINOM.DIST()
Normal Distribution	Bell-shaped curve	Excel: NORM.DIST(), NORM.INV()
Sampling Distribution	Distribution of sample statistics	Excel: Simulation tools

Additional info: Excel functions are commonly used for calculations in business statistics. Understanding both manual and software-based methods is important for exam preparation.