Hypothesized population

The first stopping criterion is straightforward, it is simply the standard hypothesis test, based on a probabilities that we have previously discussed of a sample coming from the hypothesized population P0 [2], The second stopping criterion, however, seems to fly in the face of our previous discussions on the topic, where we said that you could not prove two populations the same. 

Continuing from our previous discussion in Chapter 18 from reference [1], analogous to making what we have called (and is the standard statistical terminology) the a error when the data is above the critical value but is really from P0, this new error is called the [3 error, and the corresponding probability is called the (3 probability. As a caveat, we must note that the correct value of [3 can be obtained only subject to the usual considerations of all statistical calculations errors are random and independent, and so on. In addition, since we do not really know the characteristics of the alternate population, we must make additional assumptions. One of these assumptions is that the standard deviation of the alternate population (Pa) is the same as that of the hypothesized population (P0), regardless of the value of its mean. 

The existence of the [3 probability provides us with the tool for determining what is called the power of the test, which is just 1 - j8, the probability of coming to the correct conclusion when in fact the data did not come from the hypothesized population P0. This is the answer to our earlier question once we have defined the alternate population Pa, we can determine the /3 probability of a sample having come from Pa, just as we can determine the a probability of that sample having come from P0. 

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. A common procedure is to set up a null hypothesis, denoted by which states that there is no significant difference between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs from a null hypothesis is called an alternative hypothesis, denoted by Tfj. 

Let us digress a moment and consider when a two-tailed test is needed, and what a one-tailed test implies. We assume that the measurements can be described by the curve shown in Fig. 2.10. If so, then 95% of the time a sample from the specified population will fall within the indicated range and 5% of the time it will fall outside 2.5% of the time it is outside on the high side of the range, and 2.5% of the time it is below the low side of the range. Our assumption implies that if p does not equal the hypothesized value, the probability of its being above the hypothesized value is equal to the probability of its being below the hypothesized value. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

A one-tailed test is required since the alternative hypothesis states that the population parameter is equal to or less than the hypothesized value. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

Figure 1.34. Alternative hypothesis and the power of a t-test. Alpha (a) is the probability of rejecting an event that belongs to the population associated with it is normally in the range 0.05. .. 0.01. Beta (/3) is the probability that an event that is effectively to be associated with H is accepted as belonging to the population associated with Hq. Note that the power of the test to discriminate between hypotheses increases with the distance between ha and hb- >-a is fixed either by theory or by previous measurements, while hb can be adjusted (shifted along the x-axis), for examples see H - H4, Section 4.1. Compare with program HYPOTHESIS.

More controversially, endocrine disruption as a consequence of exposure to the herbicide atrazine (2-chloro-4-ethylamine-6-isopropylamine-s-triazine), one of the most widely used herbicides in the world, has also been hypothesized to explain various adverse biological effects in frog populations in the United States. Exposure to atrazine in the laboratory at high concentrations, far exceeding those found in the natural environment, has been reported to induce external deformities in the anuran species Rana pipiens, Rana sylvatica, and Bufo americanus (Allran and Karasov 2001). Studies by Hayes et al. have suggested that atrazine can induce hermaphroditism in amphibians at environmentally relevant concentrations (Hayes et al. 2002 Hayes et al. 2003). Laboratory studies with atrazine also indicated the herbicide... 

In clinical trial analyses you may want to test the mean for a single population to determine if that value differs from a hypothesized value. For example, let s say that you have the lab test value LDL and you want to know if the change-from-baseline value is significantly different from zero. There are several ways to perform this test in SAS. If you assume the change from baseline for LDL, ldl change, is normally distributed, you can run a one-sample t-test in SAS like this ... 

These numbers do not provide an argument for desalination on anything like the hypothesized scale, much less an argument for nuclear energy per se. But they provide an illustration of the ways in which having ample energy supplies can help to ease the support of a larger world population. 

An unanswered question about adenosine is how this inhibitory neurotransmitter activates the ventrolateral preoptic area of the hypothalamus (VLPO), which contains a population of sleep-active neurons and is hypothesized to be... 

The limits of the allowable values around the hypothesized values close in on it as n increases. This behavior is shown in Figure 20-1. If, in fact, we were to plot the mean of the population as a function of n, it would be a horizontal line, just as shown. The mean of the actual data would vary around this horizontal line (assuming the null hypothesis was correct), at smaller and smaller distances, as n increased. 

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