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