Study guide: Statistical Data Analysis

📋 Course Outline

1. Populations and Variable Types
2. Descriptive Statistics and Dispersion
3. Random Variables and Distributions
4. Normal Distributions and Z-Scores
5. Confidence Intervals and Proportions
6. Means and Student t-Intervals
7. Hypothesis Tests and Errors
8. Comparing Two Means

📖 1. Populations and Variable Types

🔑 Key Concepts & Definitions

• Population: Set of individuals or objects being studied.
• Sample: Subset of the population.
• Statistical variable: Characteristic or property of interest of a population.

★ Must-know

📌 Variables are categorical (nominal or ordinal) or numerical, with numerical variables classified as discrete or continuous.

Further detail

📌 A discrete numerical variable has a finite or countably infinite range, whereas a continuous variable is modeled as having an uncountable range.

💡 Memory Hook

A population is the full group; a sample is only a subset.

📖 2. Descriptive Statistics and Dispersion

🔑 Key Concepts & Definitions

• Median: Middle data point after the observations are ordered from smallest to largest.
• Variance: Expected value of squared deviations from the mean, measuring dispersion around that mean.
• Mode: Most frequent value or observation in a data set.
• Range: Maximum value minus the minimum value.
• Interquartile range: Third quartile minus the first quartile.

📝 Essential Points

📌 The population mean is denoted by μ, while the sample mean is denoted by x̄.

📖 3. Random Variables and Distributions

🔑 Key Concepts & Definitions

• Sample space: Set of all possible outcomes of a random experiment.
• Random variable: Numerical function assigning a number to each possible outcome of a random experiment.
• Probability mass function: Function giving the probability that a discrete random variable takes each possible value in its range.
• Expected value: Probability-weighted average of its possible values.
• Cumulative distribution function: The cumulative distribution function is denoted F(α), but the course excerpt does not provide its definition.

📖 4. Normal Distributions and Z-Scores

🔑 Key Concepts & Definitions

• Standardized value: Number of standard deviations an observation lies above or below the mean.
• Standard deviation: Square root of the variance, measuring dispersion around the expected value.

★ Must-know

• A normal distribution is bell-shaped, and its mean μ determines the position of the curve on the horizontal axis.

• For a normal distribution with unchanged mean, increasing the variance makes the curve flatter and more spread out.

Further detail

• A standardized value of Z = −2 means that the corresponding observation is two standard deviations below the mean.

💡 Memory Hook

Picture a bell curve: the mean shifts its center, while a larger standard deviation flattens and widens it.

📖 5. Confidence Intervals and Proportions

🔑 Key Concepts & Definitions

• Point estimate: The observed estimate of a population proportion π is denoted probs; observing 8 fraudulent transactions in 200 gives probs = 8/200 = 0.04.

★ Must-know

📌 A 95% confidence procedure produces intervals that contain the true parameter in approximately 95% of repeated samples; it does not mean that a particular computed interval has a 95% probability of containing the parameter.

Further detail

• For the same sample, a 99% confidence interval is approximately 31% wider and less precise than a 95% confidence interval.

• If the sample size is multiplied by 4 while π remains unchanged, the standard error of the sample proportion is divided by 2.

• For a sufficiently large random sample, the sample proportion is approximately described by a normal distribution.

📖 6. Means and Student t-Intervals

★ Must-know

• The sample variance divides the sum of squared deviations from the sample mean by n − 1, making it an unbiased estimator of the population variance σ².

📌 Student's t-distributions are bell-shaped around zero and have heavier tails than the standard normal distribution, so their central 95% interval is wider.

📌 The classical t-interval is exact for an independent random sample from a normally distributed population.

Further detail

• Because deviations from the sample mean sum to zero, only n − 1 deviations can vary independently, giving the sample variance n − 1 degrees of freedom.

• As its degrees of freedom increase, the t-distribution approaches the standard normal distribution.

• A t-based confidence interval is generally wider than the corresponding z-based interval because the t-distribution's heavier tails produce a larger critical value.

💡 Memory Hook

The z-interval uses a z-value; when population SD is unknown, the mean interval uses a t-value.

📖 7. Hypothesis Tests and Errors

🔑 Key Concepts & Definitions

• Null hypothesis: Default statement rejected only when the sample provides sufficient evidence against it.
• Alternative hypothesis: Competing research claim to which the burden of proof is assigned.
• Significance level: Maximum tolerated probability of rejecting H₀ when H₀ is true.
• p-value: Probability, assuming H₀ is true, of observing a test statistic at least as extreme as the observed statistic in the direction or directions specified by Hₐ.

📝 Essential Points

📌 A Type I error is erroneously rejecting a true H₀ (a false positive), whereas a Type II error is erroneously failing to reject H₀ (a false negative).

📌 A test rejects H₀ when its p-value is less than or equal to α.

💡 Memory Hook

Assume H₀, measure the sample evidence, then reject H₀ only if p ≤ α.

📖 8. Comparing Two Means

🔑 Key Concepts & Definitions

• Average treatment effect: In randomized controlled experiments, μtreatment − μcontrol is called the average treatment effect.

★ Must-know

• A 95% confidence interval for μ₁ − μ₂ of [0.34, 3.71] is entirely positive and provides evidence that μ₁ > μ₂.

Further detail

• An observed test statistic tobs = 2.37 means that the observed difference is 2.37 estimated standard errors above zero.

• For two means with unknown and potentially unequal variances, the standard error for their difference is based on the separate sample variances and sample sizes.

• A 95% confidence interval of [$5.84, $17.62] means the true average treatment effect is estimated to be an increase of about $5.84 to $17.62 in willingness to pay.