Type II error is an arithmetic term used within the context of hypothesis testing that illustrates the error rate which occurs when one accepts a null hypothesis that is actually false. The null hypothesis, is not rejected when it is false. Type II errors arise frequently when the sample sizes are too small and it is also called as errors of the second kind.

Suppose the mean weight of King Penguins found in an Antarctic colony last year was 5.2 kg. Assume the actual mean population weight is 5.4 kg, and the population standard deviation is 0.6 kg. At .05 significance level, what is the probability of having type II error for a sample size of 9 penguins?

H_{0} (μ_{0}) = 5.2, H_{A} (μ_{A}) = 5.4, σ = 0.6, n = 9

Beta or Type II Error rate

Let us first calculate the value of c, Substitute the values of H0, HA, σ and n in the formula,

c - μ_{0} / (σ / √n) | = -1.645 |

c - 5.2 / (0.6 / √(9)) | = -1.645 |

c - 5.2 | = -0.329 |

c | = 4.87 |

In the formula, take β to the left hand side and the other values to right hand side,
β = 1 - p(z > (c - μ_{A} / (σ / √n))) [ z = x̄ - μ_{A} / (σ / √n) ]
Substitute the values in the above equation,

β | = 1 - p(z > (4.87 - 5.4 / (0.6 / √(9)))) |

= 1 - p(z > -2.65) | |

= 1 - 0.9960 | |

= 0.0040 |

Hence the Type II Error rate value is calculated.