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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 4. Statistics & Probability / 4.12 Further Hypothesis Testing / 4.12.2 Hypothesis Testing for Mean (Two Sample)


4.12.2 Hypothesis Testing for Mean (Two Sample)


Two-Sample Tests

What is a two-sample test?

  • A two-sample test is used to compare the means (μ1 & μ2) of two normally distributed populations
    • You use a z-test when the population variances (sigma subscript 1 superscript 2 & sigma subscript 2 superscript 2) are known
    • You use a t-test when the population variances are unknown
      • In this course you will assume the variances are equal and use a pooled sample for a t-test
      • In a pooled sample the data from both samples are used to estimate the population variance

What are the steps for performing a two-sample test on my GDC?

  • STEP 1: Write the hypotheses
    • H0 : μ1= μ2
      • Clearly state that μ1 & μ2 represent the population means
      • Make sure you make it clear which mean corresponds to which population
      • In words this means that the two population means are equal
    • For a one-tailed test H1 : μ1 < μ2 or H1 : μ1 > μ2
    • For a two-tailed test: H1 : μ1 ≠ μ2
      • The alternative hypothesis will depend on what is being tested
  • STEP 2: Decide if it is a z-test or a t-test
    • If the populations variances are known then use a z-test
    • If the populations variances are unknown then use a t-test
      • Assume the variances are equal and use a pooled sample
  • STEP 3: Enter the data into your GDC and choose two-sample z-test or two-sample t-test
    • If you have the raw data
      • Enter the data as a list
      • Enter the values of σ1 & σ2 if a z-test
      • Choose the pooled option if a t-test
    • If you have summary statistics (only for a z-test)
      • Enter the values of x with bar on top subscript 1, x with bar on top subscript 2 , σ1, σ2, n1 & n2
    • Your GDC will give you the p-value
  • STEP 4: Decide whether there is evidence to reject the null hypothesis
    • If the p-value < significance level then reject H0
  • STEP 5: Write your conclusion
    • If you reject H0­ then there is evidence to suggest that...
      • The mean of the 1st population is smaller (for H1 : μ1 < μ2)
      • The mean of the 1st population is bigger (for H1 : μ1 > μ2)
      • The means of the two populations are different (for H1 : μ1 ≠ μ2)
    • If you accept H­0 then there is insufficient evidence to reject the null hypothesis which suggests that...
      • The mean of the 1st population is not smaller (for H1 : μ1 < μ2)
      • The mean of the 1st population is not bigger (for H1 : μ1 > μ2)
      • The means of the two populations are not different (for H1 : μ1 ≠ μ2)

Worked Example

The times (in minutes) for children and adults to complete a puzzle are recorded below.

Children

3.1

2.7

3.5

3.1

2.9

3.2

3.0

2.9

 

Adults

3.1

3.6

3.5

3.6

2.9

3.6

3.4

3.6

3.7

3.0

The creator of the puzzle claims children are generally faster at solving the puzzle than adults. A t-test is to be performed at a 1% significance level.

a)
Write down the null and alternative hypotheses.

4-7-4-ib-ai-sl-t-test-one-tail-a-we-solution

b)
Find the p-value for this test.

4-7-4-ib-ai-sl-t-test-one-tail-b-we-solution

c)
State whether the creator’s claim is supported by the test. Give a reason for your answer.

4-7-4-ib-ai-sl-t-test-one-tail-c-we-solution

Paired t-tests

What is a paired t-test?

  • A paired test is where you take two samples but each data point from one sample can be paired with a data point from the other sample
    • These are used when one group of members are used twice and the two results for each member are paired
      • It could be to compare the sample before and after introducing a new factor
      • It could be to compare the sample under two different conditions
  • For this test you use the differences between the pairs and treat them as one sample
    • As the variance of the differences is unlikely to be known you will use a t-test
    • For a paired test you need to assume the differences are normally distributed
      • You don’t need to assume the populations are normally distributed

What are the steps for performing a paired t-test on my GDC?

  • STEP 1: Write the hypotheses
    • H0 : μD = 0
      • Clearly state that μD represents the population mean of the differences
      • In words this means the population mean has not changed
    • For a one-tailed test H1 : μD < 0 or H1 : μD > 0
    • For a two-tailed test: H1 : μD ≠ 0
      • The alternative hypothesis will depend on what is being tested
  • STEP 2: Enter the data into your GDC and choose the one-sample t-test
    • Enter the differences as a list
      • Be consistent with the order in which you subtract paired values
    • Your GDC will give you the p-value
  • STEP 3: Decide whether there is evidence to reject the null hypothesis
    • If the p-value < significance level then reject H0
  • STEP 4: Write your conclusion
    • If you reject H0­ then there is evidence to suggest that...
      • The mean has decreased (for H1 : μD < 0)
      • The mean has increased (for H1 : μD > 0)
      • The mean has changed (for H1 : μD ≠ 0)
    • If you accept H­0 then there is insufficient evidence to reject the null which suggests that...
      • The mean has not decreased (for H1 : μD < 0)
      • The mean has not increased (for H1 : μD > 0)
      • The mean has not changed (for H1 : μD ≠ 0)

Exam Tip

  • If an exam question has two samples with the same number of members then consider carefully whether it makes sense to do a paired test or a two sample test
  • The examiner might make it look like it is a paired test to trick you!

Worked Example

In a school all students must study French and Spanish. 9 students are selected and complete a test in both subjects, the standardised scores are shown below

 

Student

1

2

3

4

5

6

7

8

9

French score

61

82

77

80

99

69

75

71

81

Spanish score

74

79

83

66

95

79

82

81

85

 

The headteacher wants to investigate whether there is a difference in the students’ scores between the two subjects. A paired t-test is to be performed at a 10% significance level.

a)
Write down the null and alternative hypotheses.

4-12-2-ib-ai-hl-paired-t-test-a-we-solution

b)
Find the p-value for this test.

4-12-2-ib-ai-hl-paired-t-test-b-we-solution

c)
Write down the conclusion to the test. Give a reason for your answer.

4-12-2-ib-ai-hl-paired-t-test-c-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 3. Geometry & Trigonometry
                                                                                        • 3.1 Geometry Toolkit
                                                                                          • 3.1.1 Coordinate Geometry
                                                                                            • 3.1.2 Radian Measure
                                                                                              • 3.1.3 Arcs & Sectors
                                                                                              • 3.2 Geometry of 3D Shapes
                                                                                                • 3.2.1 3D Coordinate Geometry
                                                                                                  • 3.2.2 Volume & Surface Area
                                                                                                  • 3.3 Trigonometry
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      Daniel Finlay

                                                                                                                                                                                                                                                                                      Author: Daniel

                                                                                                                                                                                                                                                                                      Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.


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