Data Input : In addition to α and power that are common inputs to all models, the following inputs are used

• Standard Deviation is the expected Standard Deviation of the samples to be taken. In quality control this value is usually known from past history
• Mean (accept null hypothesis) mean0: is the mean value below which a decision that the mean is null (zero) can be made. In quality control, to detect defects in production, this is the mean of departure from expected measurements that can be accepted as normal variations.
• Mean (reject null hypothesis) mean1: is the mean value above which a decision that the mean is not null (zero) can be made. In quality control, to detect defects in production, this is the mean value of departure from expected values which will trigger an alert that something is amiss.

Results Input :

• The decision borders : the decision borders.

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We are the buying department of a grocery chain, and purchases eggs from the farmers. We expect the standard deviation of the weight of eggs to be 1g, and we stipulated that the mean weight of the eggs we purchased must not be more than 1 SD (1g) less than the average of 5gs. (x = 5 - wt of egg). For every batch of egg delivered we wish to measure the eggs until we are satisfied that this bench mark is met.

Rejection Line is y = 2.7726 + 0.5 n Acceptance Line is y = -1.5581 + 0.5 n Expected sample size n = 5 Truncation sample size = 15

We set the parameters as alpha (α) = 0.05, power=0.8, expected standard deviation = 1g, bench mark for rejection of null hypothesis (eggs too small) 1 (bench mark = 1g less than mean of 5), and the bench mark for accepting the null hypothesis (eggs not too small) 0 (not less than 5 g)

The borders are as shown above and to the right. The border for rejection of null hypothesis is y = 2.7726 + 0.5 n, and for acceptance of null hypothesis -1.5581 + 0.5 n, where n = number sampled, and y = sum of x (x = 5-wt of egg). if the borders are not crossed after 15 measurements, the null hypothesis is accepted (eggs not too small)

The plot remained between the rejection and acceptance line, until the maximum number of samples was reached after 15 measurements, and at that stage the decisions to stop further measurements and accept the null hypothesis (eggs not too small) were made.

Data Input : In addition to α and power that are common inputs to all models, the following inputs are used

• Proportion (accept null hypothesis) p0: is the proportion of positives below which a decision that the proportion is null (zero) can be made. In quality control, to detect defective items in a batch of products, this is the proportion of defectives below which the batch can be accepted by the client.
• Proportion (reject null hypothesis) p1: is the proportion of positives above which a decision that the proportion is not null (zero) can be made. In quality control, to detect defective items in a batch of products, this is the proportion of defectives above which the batch will not be accepted by the client.

Results Output :

• k : or θ, the effect size log odds ratio, where k = log((p1*(1-p0)) / (p0*(1-p1)));
• The decision borders : are calculated using α, β, and k, (see references)

The transport department noticed that less than half of teenagers pass their driving test in their first attempt, a distressing and wasteful situation. A suggestion was made that a training course be offered before the test, but this requires organisation and resources, so there is a need to pre-test the course.

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It was agreed that the course should result in 80% (0.8) of the candidate to pass the driving test at the first attempt, and that it should be considered useless if the passing rate remained the current 40% (0.4). We also set our statistical parameters so that alpha = 0.05, power = 0.8.

Effect size = 1.79 Rejection border y = 1.5474 + 0.6131 n Acceptance border y = -0.8696 + 0.6131 n Max n = 18

The borders are shown to the right. The border to decide rejection of the null hypothesis (that the pass rate is 80% or better) is y = 1.5474 + 0.6131 n, and to accept the null hypothesis (that the pass rate is not better than 80%) y = -0.8696 + 0.6131 n, n being the number of candidates reviewed, and y is the cumulative number of passes.

The data is shown above and to the left, the value 1 for those who passed the test, and 0 those failed (the first, sixth, and ninth candidate).

The plot is shown to the right. By the 12th candidate, the border for rejecting the null hypothesis has been reached. The study can at this stage be terminated and the conclusion drawn that the pass rate is 80% or better.