Introduction to Inferences¶

Tails on A Normal Distribution¶

The regions on the extreme left or right sides

• For common percentages, it is easier and more accurate to use the last row of the t-table, \(\infty\)

Introduction to Hypothesis Testing¶

Hypothesis Test¶

A procedure for determining whether or not a population parameter has changed significantly from its previous parameter

Null Hypothesis¶

\(H_0\), the assumption that the population parameter has not changed

Alternative Hypothesis¶

\(H_a\), how you think the population parameter has changed

One-Tailed Test¶

When you suspect the population parameter has in a particular direction (either increase or decrease)

• \(H_a: \mu > ...\)
• \(H_a: \mu < ...\)

Two-Tailed Test¶

When you suspect the population parameter has changed

• \(H_a: \mu \neq ...\)

Test Statistics¶

An unbiased estimate of the population parameter from the recent sample from the population

Standardized Test Statistics¶

How far the test statistic is from the population parameter

\(\dfrac{statistic - parameter}{standard\ error\ of\ the\ statistic}\)

Conditions for A Hypothesis Test¶

• Independence
  1. Data is collected by random sampling or random assignment
  2. If sampling without replacement, \(n \leq 0.1 N\)
• Normality
  • Approximately symmetric
  • No outliers

p-value¶

The probability of obtaining a test statistic as extreme, or more extreme, than the on observed in the sample, assuming the null hypothesis is true

• The more extreme the test statistic is compared to the sample, the smaller the p-value, and the more evidence there is to suggest that the null hypothesis can be rejected
• \(p\)-value is calculated by finding probabilities from the sample distribution for that test statistic
  • Use the population parameter from the null hypothesis
  • Work out the standardized test statistic

Significance Level¶

\(\alpha\), the probability threshold at which you reject the null hypothesis.

\(p < \alpha \implies\) the null hypothesis should be rejected

Introduction to Confidence Intervals¶

Confidence Interval¶

A symmetric range of values centered about an estimate from a sample designed to capture the actual value of the population parameter

Confidence Level¶

The percentage of all possible confidence intervals that capture the population parameter

• \(confidence\ interval = statistic \pm margin\ of\ error\)
  • The statistic is the estimate from the sample
• \(margin\ of\ error = critical\ value \cdot standard\ error\ of\ statistic\)
  • The critical value depends on the confidence level, C%
  • The critical value is \(z\)-score
  • The standard error is an estimate of the population standard deviation from the data
  • The width of Margin of Error is half of the width of the Confidence Interval