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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 / 4.12.1 Hypothesis Testing for Mean (One Sample)


4.12.1 Hypothesis Testing for Mean (One Sample)


One-Sample z-tests

What is a one-sample z-test?

  • A one-sample z-test is used to test the mean (μ) of a normally distributed population
    • You use a z-test when the population variance (σ²) is known
  • The mean of a sample of size n is calculated x with bar on top and a normal distribution is used to test the test statistic
  • x with bar on top can be used as the test statistic
    • In this case you would use the distribution X with bar on top tilde straight N open parentheses mu comma space sigma squared over n close parentheses
      • Remember when using this distribution that the standard deviation is fraction numerator sigma over denominator square root of n end fraction
  • z equals fraction numerator x with bar on top minus mu over denominator fraction numerator sigma over denominator square root of n end fraction end fraction can be used as the test statistic
    • In this case you would use the distribution Z tilde straight N left parenthesis 0 comma 1 squared right parenthesis
      • This is a more old-fashioned approach but your GDC still might tell you the z-value when you do the test
      • You will not need to use this method in the exam as your GDC should be capable of doing the other method

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

  • STEP 1: Write the hypotheses
    • H0 : μ = μ0
      • Clearly state that μ represents the population mean
      • μ0  is the assumed population mean
    • For a one-tailed test H1 : μ < μ0 or H1 : μ > μ0
    • For a two-tailed test: H1 : μ ≠ μ0
      • The alternative hypothesis will depend on what is being tested
  • STEP 2: Enter the data into your GDC and choose the one-sample z-test
    • If you have the raw data
      • Enter the data as a list
      • Enter the value of σ
    • If you have summary statistics
      • Enter the values of x with bar on top, σ and n
    • 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 : μ < μ0)
      • The mean has increased (for H1 : μ > μ0)
      • The mean has changed (for H1 : μ ≠ μ0)
    • If you accept H­0 then there is insufficient evidence to reject the null hypothesis which suggests that...
      • The mean has not decreased (for H1 : μ < μ0)
      • The mean has not increased (for H1 : μ > μ0)
      • The mean has not changed (for H1 : μ ≠ μ0)

How do I find the p-value for a one-sample z-test using a normal distribution?

  • The p-value is determined by the test statistic x with bar on top
  • For H1 : μ < μ0 the p-value is straight P invisible function application open parentheses X with bar on top less than x with bar on top vertical line mu equals mu subscript 0 close parentheses
  • For H1 : μ > μ0 the p-value is straight P invisible function application open parentheses X with bar on top greater than x with bar on top vertical line mu equals mu subscript 0 close parentheses
  • For H1 : μ ≠ μ0 the p-value is straight P invisible function application open parentheses open vertical bar X with bar on top minus mu subscript 0 close vertical bar greater than open vertical bar x minus mu subscript 0 close vertical bar vertical line mu equals mu subscript 0 close parentheses
    • If x with bar on top less than mu subscript 0 then this can be calculated easier by 2 cross times straight P invisible function application open parentheses X with bar on top less than x with bar on top vertical line mu equals mu subscript 0 close parentheses
    • If x with bar on top greater than mu subscript 0 then this can be calculated easier by 2 cross times straight P invisible function application open parentheses X with bar on top greater than x with bar on top vertical line mu equals mu subscript 0 close parentheses

How do I find the critical value and critical region for a one-sample z-test?

  • The critical region is determined by the significance level α%
    • For H1 : μ < μ0 the critical region is X with bar on top less than c where straight P invisible function application open parentheses X with bar on top less than c vertical line mu equals mu subscript 0 close parentheses equals alpha percent sign
    • For H1 : μ > μ0 the critical region is X with bar on top greater than c where straight P invisible function application open parentheses X with bar on top greater than c vertical line mu equals mu subscript 0 close parentheses equals alpha percent sign
    • For H1 : μ ≠ μ0 the critical regions are X with bar on top less than c subscript 1 and X with bar on top greater than c subscript 2 where straight P invisible function application open parentheses X with bar on top less than c subscript 1 vertical line mu equals mu subscript 0 close parentheses equals straight P invisible function application open parentheses X with bar on top greater than c subscript 2 vertical line mu equals mu subscript 0 close parentheses equals 1 half alpha percent sign
  • The critical value(s) can be found using the inverse normal distribution function
    • When rounding the critical value(s) you should choose:
      • The lower bound for the inequalities X with bar on top less than c
      • The upper bound for the inequalities X with bar on top greater than c
    • This is so that the probability does not exceed the significance level

Exam Tip

  • Exam questions might specify a method for you to use so practise all methods (using GDC, p-values, critical regions)
  • If the exam question does not specify a method then use whichever method you want
    • Make it clear which method you are using
    • You can always use a second method as a way of checking your answer

Worked Example

The mass of a Burmese cat, C , follows a normal distribution with mean 4.2 kg and a standard deviation 1.3 kg.  Kamala, a cat breeder, claims that Burmese cats weigh more than the average if they live in a household which contains young children.  To test her claim, Kamala takes a random sample of 25 cats that live in households containing young children.

a)
State the null and alternative hypotheses to test Kamala’s claim.

4-12-1-ib-ai-hl-one-sample-z-test-a-we-solution

b)
Using a 5% level of significance, find the critical region for this test.

4-12-1-ib-ai-hl-one-sample-z-test-b-we-solution

c)
Kamala calculates the mean of the 25 cats included in her sample to be 4.65 kg. Determine the conclusion of the test.

4-12-1-ib-ai-hl-one-sample-z-test-c-we-solution

One-Sample t-tests

What is a one-sample t-test?

  • A one-sample t-test is used to test the mean (μ) of a normally distributed population
    • You use a t-test when the population variance (σ²) is unknown
    • You need to use the unbiased estimate for the population variance (s subscript n minus 1 end subscript superscript 2)
  • The mean of a sample of size n is calculated x with bar on top and a t-distribution is used to test it
    • t-distributions are similar to normal distributions
      • As the sample size gets larger the t-distribution tends towards the standard normal distribution
  • You won’t be expected to find the critical value
    • The p-value can be found using the test function on your GDC

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

  • STEP 1: Write the hypotheses
    • H0 : μ = μ0
      • Clearly state that μ represents the population mean
      • μ0  is the assumed population mean
    • For a one-tailed test H1 : μ < μ0 or H1 : μ > μ0
    • For a two-tailed test: H1 : μ ≠ μ0
      • 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
    • If you have the raw data
      • Enter the data as a list
    • If you have summary statistics
      • Enter the values of x with bar on top, sn-1 (sometimes written as sx on a GDC)   and n
    • 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 : μ < μ0)
      • The mean has increased (for H1 : μ > μ0)
      • The mean has changed (for H1 : μ ≠ μ0)
    • If you accept H­0 then there is insufficient evidence to reject the null hypothesis which suggests that...
      • The mean has not decreased (for H1 : μ < μ0)
      • The mean has not increased (for H1 : μ > μ0)
      • The mean has not changed (for H1 : μ ≠ μ0)

Worked Example

The IQ of a student at Calculus High can be modelled as a normal distribution with mean 126. The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

a)
State the null and alternative hypotheses to test the headteacher’s suspicion.

4-12-1-ib-ai-hl-one-sample-t-test-a-we-solution

b)
The headteacher selects 15 students and asks them to complete an IQ test.  The mean score for the sample is 127.1 and the sample variance is 14.7. Calculate the unbiased estimate for the population variance  s subscript n minus 1 end subscript superscript 2.

4-12-1-ib-ai-hl-one-sample-t-test-b-we-solution

c)
Calculate the p-value for the test.

4-12-1-ib-ai-hl-one-sample-t-test-c-we-solution

d)
State whether the headteacher’s suspicion is supported by the test.

4-12-1-ib-ai-hl-one-sample-t-test-d-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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