TYPE II Error: Fail to Reject Ho when Ho is False
When we evaluate the nullhypothesis, we can make 2 types of errors.
Consider p, the true proportion of voters who favor a particular political candidate. A pollster is interested in testing at the α = 0.01 level, the null hypothesis H0: p = 0.50 against the alternative hypothesis that HA: p > 0.50. Find the sample size n that is necessary to achieve 0.80 power at the alternative p = 0.55.
P(TYPE II Error) = P(Fail to Reject Ho | Ho is False) = β = beta
So, in summary, if the agricultural researcher collects data on n = 13 corn plots, and rejects his null hypothesis H0: μ = 40 if the average crop yield of the 13 plots is greater than 42.737 bushels per acre, he will have a 5% chance of committing a Type I error and a 10% chance of committing a Type II error if the population mean μ were actually 45 bushels per acre.
not rejecting the null hypothesis when the alternative is true.
Solution. Setting α, the probability of committing a Type I error, to 0.05, implies that we should reject the null hypothesis when the test statistic Z ≥ 1.645, or equivalently, when the observed sample mean is 103.29 or greater:
the null hypothesis is rejected when it is true.
Solution. Setting α, the probability of committing a Type I error, to 0.01, implies that we should reject the null hypothesis when the test statistic Z ≥ 2.326, or equivalently, when the observed sample mean is 109.304 or greater:
The failure to reject does not imply the null hypothesis is true.
Now, what would do you suppose would happen to the power of our hypothesis test if we were to change our willingness to commit a Type I error? Would the power for a given value of μ increase, decrease, or remain unchanged? Suppose, for example, that we wanted to set α = 0.01 instead of α = 0.05? Let's return to our example to explore this question.