Normal Distribution & Confidence Intervals Cheat Sheet

This cheat sheet covers key concepts and calculations related to normal distributions, including probabilities, percentiles, confidence intervals, and the Central Limit Theorem. It provides formulas and examples for practical application in statistics.

Core Principles

• Normal distribution is bell-shaped, symmetric, and defined by mean (μ) and standard deviation (σ).
• The Central Limit Theorem (CLT) states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of population distribution.
• Confidence intervals provide a range of values likely to contain the population parameter.
• Sample size significantly impacts the precision of estimates and confidence levels.

Action Steps

• Identify if the problem involves a single observation or a sample mean.
• Determine if the population standard deviation (σ) is known or unknown.
• Check if the Central Limit Theorem applies (n > 30 or normal population).
• Select the appropriate distribution (Z or t) and formula.
• Calculate Z-scores or t-scores.
• Use calculator functions (normalcdf, invNorm) or tables for probabilities and critical values.
• Construct confidence intervals using the appropriate formula.
• Calculate required sample sizes based on desired margin of error and confidence level.

Formulas

• Z-score: $Z = \frac{X - \mu}{\sigma}$
• Probability for Normal Distribution: $P(a < X < b) = \text{normalcdf}(a, b, \mu, \sigma)$
• Inverse Normal for Percentiles: $X = \text{invNorm}(p, \mu, \sigma)$
• Standard Error of the Mean: $\sigma_\bar{X} = \frac{\sigma}{\sqrt{n}}$
• Confidence Interval for Mean (known σ): $\bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
• Confidence Interval for Mean (unknown σ): $ar{X} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
• Sample Size for Mean: $n = \left(\frac{Z_{\alpha/2} \cdot \sigma}{E}\right)^2$
• Sample Size for Proportion: $n = \left(\frac{Z_{\alpha/2}}{E}\right)^2 \cdot p \cdot q$

Key Terms

• Normal Distribution: A continuous probability distribution characterized by its bell shape, symmetric around the mean.
• Mean (μ or X̄): The average value of a dataset or distribution.
• Standard Deviation (σ or s): A measure of the spread or dispersion of data around the mean.
• Z-score: The number of standard deviations a data point is from the mean.
• Central Limit Theorem (CLT): States that the sampling distribution of the sample mean approaches normality as n increases.
• Confidence Interval (CI): A range of values that is likely to contain the population parameter with a certain level of confidence.
• Confidence Level (CL): The probability that a confidence interval contains the true population parameter (e.g., 95%).
• Margin of Error (E): Half the width of the confidence interval; the maximum likely difference between the sample statistic and the population parameter.
• Percentile: The value below which a certain percentage of observations fall.

Real World Examples

• GMAT Scores: Calculating the probability of scoring above a certain threshold or finding the score needed for the top 5%.
• Pregnancy Length: Determining the proportion of pregnancies falling within a specific duration range or finding the duration for a given percentile.
• Acres Burned by Fire: Finding the probability of a certain range of acres being burned or identifying the acreage for a specific percentile.
• Movie Theater Spending: Calculating the percentage of customers spending less than a certain amount or finding the spending amount for the top percentile.
• Glass Garbage Generation: Applying CLT to find the probability of a sample mean falling within a range and verifying CLT applicability.
• SAT Scores: Comparing probabilities for individual scores versus sample means and understanding CLT application.
• Test Scores: Calculating individual probabilities, identifying scores for top percentiles, and estimating expected values for groups.
• Height Confidence Intervals: Constructing CIs for population means and understanding how sample size affects the error bound.
• Tax Form Completion Time: Calculating CIs, determining sample size for desired confidence/error, and analyzing effects of changing confidence level or sample size.
• Driver Buckling Up: Constructing CIs for population proportions and calculating the minimum sample size for a desired margin of error and confidence.

Timeline

• Early 19th Century: Development of the normal distribution (Gaussian distribution) by mathematicians like Gauss and Laplace.
• Late 19th Century: Francis Galton applies the normal distribution to biological data, coining the term 'regression towards mediocrity'.
• Early 20th Century: William Sealy Gosset (Student) develops the t-distribution for small sample sizes when population standard deviation is unknown.
• 1920s-1930s: Ronald Fisher refines statistical methods, including hypothesis testing and the use of Z and t distributions.
• Mid-20th Century: The Central Limit Theorem becomes a cornerstone of inferential statistics, enabling analysis of sample means.
• Late 20th Century - Present: Widespread use of statistical software and calculators simplifies complex calculations involving normal distributions and confidence intervals.

People

• Carl Friedrich Gauss: Developed the normal distribution, also known as the Gaussian distribution.
• Pierre-Simon Laplace: Independently developed the normal distribution and contributed to probability theory.
• Francis Galton: Applied the normal distribution to human characteristics and coined 'regression'.
• William Sealy Gosset ('Student'): Developed the t-distribution for small sample inference.
• Ronald Fisher: Pioneered many statistical methods, including hypothesis testing and ANOVA.