Sample proportions

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

The assumption in step 1 would first he tested hy obtaining a random sample. Under the assumption that p <. 02, the distrihiition for a sample proportion would he defined hy the z distrihiition. This distrihiition would define an upper hound corresponding to the upper critical value for the sample proportion. It would he unlikely that the sample proportion would rise above that value if, in fact, p <. 02. If the observed sample proportion exceeds that limit, corresponding to what would he a very unlikely chance outcome, this would lead one to question the assumption that p <. 02. That is, one would conclude that the null hypothesis is false. To test, set... 

The sample proportion is denoted r The population proportion is denoted 0... 

Construct the Lagrange multiplier statistic for testing the hypothesis that all of the slopes (but not the constant term) equal zero in the binomial logit model. Prove that the Lagrange multiplier statistic is iiR2 in the regression of (y - P) on the xs, where P is the sample proportion of ones. 

One defective item in a sample is shown to be most probable but this sample proportion occurs less than four times out of ten, even though it is the same as the population proportion. In the previous example, we would expect about one out of ten sampled items to be defective. We have intuitively taken the population proportion (p = 0.1) to be the expected value of the proportion in the random sample. This proves to be correct. It can be shown that for the binomial distribution ... 

In the example quoted earlier, we found that 42 out of a sample of 50 patients (84 per cent) showed a successful response to treatment, but, what would happen if we were to adopt this treatment and record the outcomes for thousands of patients over the next few years The proportion of successful outcomes would (hopefully) settle down to a figure in the region of 84 per cent, but it would be most surprising if our original sample provided an exact match to the long-term figure. To deal with this, we quote 95 per cent confidence intervals for the proportion in the population based upon a sample proportion. 

Pi = Xi/ui, the sample proportion from population 1 P2 = X2/ri2, the sample proportion from population 2 a = significance level Ho = null hypothesis Hi = alternative hypothesis... 

A sample splitter described by Fooks [24] consists of a feeder funnel through which the sample is fed. It passes on to the apex of two resting cones, the lower fixed and the upper adjustable by means of a spindle. Segments are cut from both cones and by rotation of the upper cone the effective area of the slits can be varied to vary the sampling proportion. Material, which falls through the segmental slots, is passed to a sample pot. The residue passes over the cone and out of the base of the unit. 

Sample Proportion P Al Sn T pH before after Crystallization time Structure... 

Sample Proportion Na Al P Sn Si T Template Crystallization temp. Structure Adsorption... 

At this point, it is worth emphasizing the difference between the terms "standard error" and "standard deviation," which, despite the same initial word, represent very different aspects of a data set. Standard error is a measure of how certain we are that the sample mean represents the population mean. Standard deviation is a measure of the dispersion of the original random variable. There is a standard error associated with any statistical estimator, including a sample proportion, the difference in two means, the difference in two proportions, and the ratio of two proportions. When presented with the term "standard error" in these applications the concept is the same. The standard error quantifies the extent to which an estimator varies over samples of the same size. As the sample size increases (for the same standard deviation) there... 

For this method we have sample proportions for independent groups 1 and 2 defined as above ... 

So the reliability factor for interval estimates will come from the Z distribution. Then a two-sided (1 - a)% confidence interval for the difference in sample proportions, p - p is ... 

As with the previous method, the first step is to calculate the point estimate, but this time the point estimate of the difference in sample proportions. For the placebo group the proportion is 0.06. For the active group the proportion... 

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