CAPE PENINSULA UNIVERSITY OF TECHNOLOGY STAT151X intermediate

STAT151X: statistics 1B

Free study notes for STAT151X (STAT 151X) — statistics 1B at CAPE PENINSULA UNIVERSITY OF TECHNOLOGY, one topic at a time, each with worked examples and a practice quiz.

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Course Syllabus

  1. Probability Distributions
  2. Sampling Distributions and Estimation
  3. Hypothesis Testing - One Sample
  4. Hypothesis Testing - Two Samples
  5. Simple Linear Regression and Correlation
  6. Chi-Square Tests and Non-Parametric Methods

Study Notes

Probability Distributions

Probability distributions are formal mathematical descriptions of the likelihood of different outcomes for a random variable. They are classified into discrete and continuous types based on the nature of their random variable's support.

A discrete random variable $X$ has a countable number of possible outcomes. Its probability distribution is defined by a Probability Mass Function (PMF), $P(X=x)$, where $x$ is a specific outcome.

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Sampling Distributions and Estimation

  • Expected Value of the Sample Mean:
    $E[\bar{X}] = E\left[\frac{1}{n} \sum_{i=1}^{n} X_i\right] = \frac{1}{n} \sum_{i=1}^{n} E[X_i] = \frac{1}{n} \sum_{i=1}^{n} \mu = \frac{n\mu}{n} = \mu$
    This demonstrates that $\bar{X}$ is an unbiased estimator of $\mu$.
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Hypothesis Testing - One Sample

Hypothesis testing, at its core, is a formal procedure for making an informed decision about a population parameter based on sample data. The framework begins with the formulation of two competing hypotheses: the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$).

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Hypothesis Testing - Two Samples

Hypothesis testing for two samples primarily involves comparing parameters (means, proportions, variances) from two distinct populations based on sample data. The fundamental principle remains the construction of a null hypothesis ($H_0$), representing no difference, and an alternative hypothesis ($H_1$ or $H_a$), representing a significant difference.

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Simple Linear Regression and Correlation

Simple Linear Regression (SLR) models the relationship between two continuous quantitative variables: a dependent variable, $Y$, and an independent variable, $X$. This relationship is assumed to be linear in its parameters. Correlation quantifies the strength and direction of the linear association between these variables.

To find $\hat{\beta}_0$ and $\hat{\beta}_1$, we take partial derivatives of $SSR$ with respect to $\hat{\beta}_0$ and $\hat{\beta}_1$, set them to zero, and solve the resulting system of "normal equations."

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STAT151X: common questions

What does STAT151X cover?

This STAT151X study guide covers 6 topics: Probability Distributions, Sampling Distributions and Estimation, Hypothesis Testing - One Sample, Hypothesis Testing - Two Samples, Simple Linear Regression and Correlation, Chi-Square Tests and Non-Parametric Methods.

Are these STAT151X notes free?

Yes. Every STAT151X note on this page is free to read without an account. A free account adds practice quizzes, flashcards and a revision plan.

How do I revise for my STAT151X exam?

Read the STAT151X note for each topic, then take its practice quiz: the questions you get wrong come back as flashcards until you know them. Save the course to get a revision plan built around your exam date.