Probabilities of Errors (College Board AP® Statistics)

Study Guide

Written by: Mark Curtis

Reviewed by: Dan Finlay

Probability of a Type I error

How do I find the probability of a Type I error if a significance level is given?

If the significance level of a test, $α$ , is known, then
- the probability of a Type I error is equal to the significance level, $α$
This is because the probability of a Type I error is
- the probability of rejecting $H_{0}$ given that $H_{0}$ is true
- which is the same definition as a significance level, $α$
  - $α$ is the preset probability at which you decide to reject $H_{0}$ , assuming it was true

Worked Example

A z-test for the population proportion, $p$ , at the significance level $α = 0.05$ has the hypotheses:

$H_{0} : p = 0.6 H_{a} : p > 0.6$

Find the probability of a Type I error.

Answer:

The significance level is given, so the P(Type I error) $= α$

The probability of a Type I error is 0.05

How do I find the probability of a Type I error if a critical region is given?

If a significance level is not given, you may be given a critical region instead
- This is the range of values for the test statistic for which the null hypothesis is rejected
- The probability of a Type I error is the probability of being in the critical region (rejecting H₀) given H₀was true

Worked Example

A z-test for the population proportion, $p$ , has the hypotheses:

$H_{0} : p = 0.6 H_{a} : p > 0.6$

The null hypothesis will be rejected if a random sample of size 50 taken from the population has a sample proportion, $\hat{p}$ , greater than 0.7.

Find the probability of a Type I error.

Answer:

The critical region is $\hat{p} > 0.7$

The significance level is not given, so the P(Type I error) = P(in the critical region, given that the null hypothesis is true)

$P (\hat{p} > 0.7 given that p = 0.6)$

Calculate the standardized test statistic $z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$ for $\hat{p} = 0.7$ where $p_{0} = 0.6$ and $n = 50$

$z = \frac{0.7 - 0.6}{\sqrt{\frac{0.6 (1 - 0.6)}{50}}} = 1.4433756 . . .$

Find the probability that $\hat{p} > 0.7$ , i.e. $P (Z > 1.4433756 . . .)$ , for example using the z-tables

$1 - 0.9251 = 0.0749$

The probability of a Type I error is 0.0749

Probability of a Type II error

How do I find the probability of a Type II error if a critical region is given?

Recall that a critical region is the range of values for the test statistic for which the null hypothesis is rejected
The probability of a Type II error is the probability of not rejecting the null hypothesis, despite it being false in reality
- This is the probability of not being in the critical region (not rejecting H₀) given H₀was false
  - You need to be given the actual (true) population parameter to find this
  - For example, H₀ assumed $p = \frac{1}{2}$ but actually $p = \frac{1}{3}$
  - P(Type II error) = P(not in the critical region, given the actual population parameter is true)

How do I reduce the probability of a Type II error?

The probability of a Type II error is reduced when one of the following is changed (and the others are kept the same):
- The sample size, $n$ , increases
- The significance level, $α$ , increases
- The standard error of the hypothesis test decreases
- The actual (true) population parameter is farther from the null population parameter

Because $α$ is also the probability of a Type I error, the above means:
- You can reduce the probability of a Type II error by increasing the significance level
  - However this will increase the probability of a Type I error
- You can reduce the probability of a Type I error by reducing the significance level
  - However this will increase the probability of a Type II error
- The only way to reduce both probabilities is by increasing the size of the sample

Worked Example

A z-test for the population proportion, $p$ , has the hypotheses:

$H_{0} : p = 0.6 H_{a} : p > 0.6$

The null hypothesis will be rejected if a random sample of size 50 taken from the population has a sample proportion, $\hat{p}$ , greater than 0.7.

Given that, in reality, the population proportion is actually $p = 0.75$ , find the probability of a Type II error.

Answer:

The critical region is $\hat{p} > 0.7$ , so not being in the critical region is $\hat{p} < 0.7$

P(Type II error) = P(not in the critical region, given the actual population parameter is true)

$P (\hat{p} < 0.7 given that p = 0.75)$

Calculate the standardized test statistic $z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$ for $\hat{p} = 0.7$ where $p_{0} = 0.75$ and $n = 50$

$z = \frac{0.7 - 0.75}{\sqrt{\frac{0.6 (1 - 0.6)}{50}}} = - 0.721687 . . .$

Find the probability that $\hat{p} < 0.7$ , i.e. $P (Z < - 0.721687 . . .)$ , for example using the z-tables

0.2358

The probability of a Type II error is 0.2358

Last updated: 4 September 2024

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Previous:Type I & Type II ErrorsNext:Power of a Test

Probabilities of Errors (College Board AP® Statistics)

Study Guide

Probability of a Type I error

How do I find the probability of a Type I error if a significance level is given?

How do I find the probability of a Type I error if a critical region is given?

Probability of a Type II error

How do I find the probability of a Type II error if a critical region is given?

How do I reduce the probability of a Type II error?

How are the probabilities of a Type I error and a Type II error related?

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Unit 1: Exploring One-Variable Data

Summary Statistics

Describing Variables

Parameters & Statistics

Measures of Center

Measures of Position

Measures of Variability

Tables & Relative Frequency

Grouped Data

Outliers & Resistant Measures

Five-Number Summary & Boxplots

Skewness of Data

Comparing Data using Summary Statistics

Graphical Representations

Shape of Distributions

Bar Charts & Histograms

Dotplots & Stemplots

Cumulative Graphs

Comparing Univariate Graphs

The Normal Distribution

Properties of Normal Distributions

Standardized z-scores

Comparing Normal Distributions

Finding Proportions from Normal Distributions

Inverse Normal Calculations

Estimating Parameters of Normal Distributions

Unit 2: Exploring Two-Variable Data

Tables & Graphs

Two-Way Tables & Relative Frequencies

Bar Graphs & Mosaic Plots

Scatterplots & Regression

Explanatory & Response Variables

Scatterplots

Association & Correlation Coefficients

Interpolation & Extrapolation using Linear Models

Residuals

The Least-Squares Regression Line

Residual Plots

The Coefficient of Determination

Outliers, High-Leverage & Influential Points

Linearization of Bivariate Data

Unit 3: Collecting Data

Sampling Methods & Bias

Introduction to Sampling

Simple Random Sampling (SRS)

Random Sampling Methods

Types of Bias

Non-random (Biased) Sampling Methods

Experimental Design

Introduction to Experiments

Well-Designed Experiments

Control Groups, Placebos & Blind Experiments

Completely Randomized Design

Randomized Block & Matched Pairs Design

Unit 4: Probability, Random Variables & Probability Distributions

Probability

Estimating Probability using Relative Frequency

Probabilities of Single Events

Introduction to Combined Events

Addition Rule & Mutually Exclusive Events

Conditional Probability

Multiplication Rule & Independent Events

Probabilities of Combined Events using Tree Diagrams

Probabilities of Combined Events using the Rules

Discrete Random Variables

Probability Distributions for Discrete Random Variables

Cumulative Probability Distributions for Discrete Random Variables

Mean & Standard Deviation of a Discrete Random Variable