## Exam-Style Question on Significance## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 463. This question is similar to one that appeared on an IB AI Standard paper (specimen) for 2021. The use of a calculator is allowed.

The Scrumptious Sweet Company sell a variety pack of colourful, shaped sweets. The sweets are produced such that 60% are square and 40% are circular. It is known that 20% of the square shaped sweets and 40% of the circular sweets are coloured red.

(a) Show this information in a tree diagram.

A sweet is selected at random.

(b) Find the probability that the sweet is red.

(c) Given that the sweet is red, find the probability it is circular.

The Scrumptious Sweet Company also produce variety packs of Rainbow Gums. Their specifications state that the colours in each variety pack should be distributed as follows.

Colour | Red | Orange | Yellow | Green | Blue |

Percentage (%) | 15 | 25 | 15 | 25 | 20 |

Inspector Lou Spowels opens a pack of 90 sweets and records the frequency of each colour:

Colour | Red | Orange | Yellow | Green | Blue |

Observed Frequency | 12 | 21 | 15 | 20 | 22 |

To investigate if the sample is consistent with the company's specifications, Mr Spowels conducts a \(\chi^2\) goodness of fit test. The test is carried out at a 5% significance level.

(d) Write down the null hypothesis for this test.

(e) Copy and complete the following table giving the frequencies correct to one decimal place:

Colour | Red | Orange | Yellow | Green | Blue |

Expected Frequency |

(f) Write down the number of degrees of freedom.

(g) Find the p-value for the test.

(h) State the conclusion of the test. Give a reason for your answer.

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If you need more practice try the self-checking interactive exercises called Tree Diagrams. There is also a slide presentation you might find useful on the Chi squared GOF test.

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