Hypothesis Formulation in Research: Types and Uses

Hypothesis Formulation

Scenarios show us what we are looking for or trying to prove and can be defined as tentative explanations of the phenomenon under investigation, made by way of propositions. A variable is a property that may vary (to acquire different values) and whose variation is susceptible to measurement.

Research: Hypotheses, Questions, and Objectives

Assumptions commonly arise from the objectives and research questions.

Origin of Hypotheses

Our hypotheses can arise from a postulate of a theory,

Read More

Correlation, Probability, and Hypothesis Testing

Understanding Correlation and Its Applications

1. Types of Correlations:

  • Positive: Both variables move in the same direction.
  • Negative: Variables move in opposite directions.
  • Zero: No relationship between the two variables.

2. Scatterplots: Visual representations of the relationship between two variables.

3. Correlation Scale: Ranges from -1 to 0 (negative correlation) and from 0 to +1 (positive correlation).

4. Formulas: Include formulas for covariance and the correlation coefficient in your cheat sheet.

Read More

Understanding Exponents, Divisibility, and Fractions

Understanding Exponents

Exponentiation is a multiplication of equal factors. For example: 2.2.2.2.2 = 25 (where 5 is the exponent and 2 is the base). The base indicates how many times it is multiplied by itself.

  • 43 = 4.4.4 = 64
  • 8.8.8.8.8 = 85
  • Base 2, exponent 6 = 64
  • Base 0, exponent 9 = 0

Order of operations with exponents:

  1. Powers
  2. Multiplications
  3. Additions and subtractions

For example: 3.24 + 25 = 3.16 (= 24) + 32 (= 25) = 48 + 32 = 80.

Properties of Exponentiation

  • Multiplication of powers with the same base:
Read More

Statistics and ggplot2: Quick Reference

Statistics and ggplot2: Quick Reference

Central Tendency

  • Mean: The average of values, affected by outliers.
    • Formula: \(\bar{x} = \frac{\Sigma x_i}{n}\)
  • Median: The middle value, robust to outliers.
  • Mode: The most frequent value in a dataset.

Variability Metrics

  • Range: \(\text{Max} – \text{Min}\)
  • Population Variance: \(\sigma^2 = \frac{\Sigma (x_i – \mu)^2}{N}\)
  • Sample Variance: \(s^2 = \frac{\Sigma (x_i – \bar{x})^2}{n-1}\) (Bessel’s correction).
  • Standard Deviation (SD): The square root of variance.
    • Formula:
Read More

Statistical Inference: Z-Distribution, T-Distribution, and Regression

Chapter 6: Standard Error (SE)

The standard error (SE) is the standard deviation of the sampling distribution of a statistic. It measures the precision of the sample statistic as an estimate of the population parameter. A z-distribution is the standard normal distribution with a mean of 0 and a standard deviation of 1. It is used for testing hypotheses about a single population mean or proportions when σ is known. The T-distribution is a family of distributions that are similar to the normal distribution

Read More

Vehicle Tax Analysis: Price, Age, and Regression Insights

1. Interpreting the Slope in the Simple Linear Regression Model

A 1% increase in price is associated with a 0.8% increase in taxes. Given that the increase is less than 1%, the vehicle tax is regressive, not progressive, meaning that more expensive cars pay proportionally less tax.

rate = exp(b1) * exp(0.8161 * log_price) = exp(b1) * (exp(log_price))^0.8161 = exp(b1) * (price)^0.8161

Hence, an increase in the price of 1% implies an increase in the rate of (1.01)^0.8161 = 1.00815, that is, an increase

Read More