# wefail to reject H0 and do not accept H1

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

## 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.

### One can never prove the truth of a statistical (null) hypothesis.

As this plot suggests, if we are interested in increasing our chance of rejecting the null hypothesis when the alternative hypothesis is true, we can do so by increasing our sample size n. This benefit is perhaps even greatest for values of the mean that are close to the value of the mean assumed under the null hypothesis. Let's take a look at two examples that illustrate the kind of sample size calculation we can make to ensure our hypothesis test has sufficient power.

### failing to reject the null hypothesis when it is true.

And, the probability of rejecting the null hypothesis at the α = 0.05 level when μ = 116 is greater than 0.999999, as calculated here:

### rejecting the null hypothesis when it is false.

Again, we start by finding a threshold value c, such that if the observed sample proportion is larger than c, we'll reject the null hypothesis: