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These are the Transum resources related to the statement: "Formulation of null and alternative hypotheses, H_{0} and H_{1}. Significance levels. p-values. Expected and observed frequencies. The Χ^{2} test for independence: contingency tables, degrees of freedom, critical value. The Χ^{2} goodness of fit test. The τ-test. Use of the p-value to compare the means of two populations. Using one-tailed and two-tailed tests".

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

- First Impressions This activity will collect data about your first impressions of some optical illusions. You can then analyse the data to come to your own conclusions.
- Significance A slide presentation showing how to use the chi-squared test to measure significance.
- Statistical Significance This video explains numeric and categorical data and helps you gain an understanding of when to apply a t-test or a chi-square test.
- t-Test Revision A slide presentation designed to revise the key aspects of Student's t-Test.

Here are some exam-style questions on this statement:

- "
*A group of students were asked how they get to school each day. The results are shown in the following table*" ... more - "
*One hundred and sixty students are asked how many portions of fruit or vegetables on average they have per day. The data collected is organized in the following table.*" ... more - "
*Ucayali, a multinational technology company that focuses on e-commerce, has a distribution centre in Camelot. As part of a study into healthy lifestyles, the number of cups of coffee drunk by 420 staff members each day were recorded. Results are shown in the table.*" ... more - "
*The older students from Glee High School are required to follow a two year IB Mathematics course. Data were gathered from a sample of 242 students regarding their choice of course. The following data were recorded.*" ... more - "
*The staff of a shop kept records of who bought smart phones during the month of February one year. They looked at the numbers of phones bought by gender and the size of the screens. This information is shown in the table below; S represents the size of the screen in centimetres.*" ... more - "
*The 2nd Rutherford American Scouts joined the 37th Wolverhampton British Scouts for an International Camp. Skipper Jones is interested to see if the mean height of American Scouts, \( \mu_1\), is the same as the mean height of British Scouts, \( \mu_2\). The information is recorded in the following table.*" ... more - "
*A survey was conducted asking people's opinions on whether mouthwash should be used before or after brushing teeth. The results are shown as follows.*" ... more

Click on a topic below for suggested lesson starters, resources and activities from Transum.

- Probability Probability is a measure of the weight of evidence, and is arrived at through reasoning and inference. In simple terms it is a measure or estimation of likelihood of the occurrence of an event. The word probability comes from the Latin word probabilitas which is a measure of the authority of a witness in a legal case. Some of the earlier mathematical studies of probability were motivated by the desire to be more profitable when gambling. Today however the practical uses of probability theory go far beyond gambling and are used in many aspects of modern life. We believe that even adults can, in many cases, have a poor intuition regarding the effects of probability. These activities are designed to help pupils calculate but also get a 'feel' for the principles of probability.
- Statistics Statistics is the study of the collection, organisation, analysis, interpretation and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. It also includes describing mathematical relationships between variables and presenting these to an audience in a way that best conveys meaning. See also the topics called Data Handling, Probability and Averages.

The IB AI Syllabus provides the following guidance and clarification:

Students should express H_{0} and H_{1} as an equation or inequality, or in words as appropriate.

- the maximum number of rows or columns in a contingency table will be 4
- the degrees of freedom will always be greater than one. At SL the degrees of freedom for the goodness of fit test will always be n…1
- the 𝜒
^{2}critical value will be given if appropriate - students will be expected to use technology to find a p -value and the 𝜒
^{2}statistic - only questions on upper tail tests with commonly-used significance levels (1%, 5%, 10%) will be set
- students will be expected to either compare a p -value to the given significance level or compare the 𝜒
^{2}statistic to a given critical value - expected frequencies will be greater than 5.

Hand calculations of the expected values or the 𝜒^{2} statistic may enhance understanding.

If using 𝜒^{2} tests in the IA, students should be aware of the limitations of the test for expected frequencies of 5 or less.

In examinations calculations will be made using technology.

At Standard Level, samples will be unpaired, and population variance will always be unknown.

Students will be asked to interpret the results of a test.

Students should know that the underlying distribution of the variables must be normal for the t-test to be applied. In examinations, students should assume that variance of the two groups is equal and therefore the pooled two-sample t-test should be used.

How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.