Question

Mars Inc. claims that they produce M&Ms with the following
distributions:

Brown | 30% | Red | 20% | Yellow | 20% |

Orange | 10% | Green | 10% | Blue | 10% |

A bag of M&Ms was randomly selected from the grocery store
shelf, and the color counts were:

Brown | 23 | Red | 23 | Yellow | 20 |

Orange | 15 | Green | 17 | Blue | 15 |

Using the χ^{2} goodness of fit test to determine if the
proportion of M&Ms is what is claimed, what is the test
statistic?

a) χ^{2} = 10.409

b) χ^{2} = 1.960

c) χ^{2} = 9.109

d) χ^{2} = 7.209

e) χ^{2} = 18.218

f) None of the above

Answer #1

Here we have to test the hypothesis that,

H0 : All the population proportions are equal.

H1 : Atlease one population proportion differ.

Assume alpahaa = 0.05

The test statistic follows X^{2} distribution with k-1
degrees of freedoms.

where k = no. of groups= 6

The test statistic is,

where O is observed frequency

E is expected frequency.

We have given proportions and O.

Now we can find E by using formula,

E = N*p

where N = 113

colour | p | E | O | (O-E)^2/E |

brown | 0.3 | 33.9 | 23 | 3.50472 |

red | 0.2 | 22.6 | 23 | 0.00708 |

yellow | 0.2 | 22.6 | 20 | 0.299115 |

orange | 0.1 | 11.3 | 15 | 1.211504 |

green | 0.1 | 11.3 | 17 | 2.875221 |

blue | 0.1 | 11.3 | 15 | 1.211504 |

total | 113 | 113 | 9.109145 |

Test statistic = 9.109

Option c) is correct.

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