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R = Pr [X + z > Y}.

You might like to use the Kolmogorov-Smirnov Test for Two Populations and Comparing Two Random Variables in checking your computations and performing some numerical experiment for a deeper understanding of these concepts.

Further Readings:
Arsham H., A generalized confidence region for stress-strength reliability, IEEE Transactions on Reliability, 35(4), 586-589, 1986.
Conover W., Practical Nonparametric Statistics, Wiley, 1998.
Hollander M., and D. Wolfe, Nonparametric Statistical Methods, Wiley, 1999.
Kotz S., Y. Lumelskii, and M. Pensky, The Stress-Strength Model and Its Generalizations: Theory and Applications,
Imperial College Press, London, UK, 2003, distributed by World Scientific Publishing.

Hypotheses Testing

Let us consider a simple problem of inference about population mean. We have a large population with known mean. We take a sample and wish to know whether the sample mean is significantly different from the population mean. Our null hypothesis is that it is not.

The theory of probability is only capable of dealing with random variables which generate a frequency distribution "in the long run". We have one fixed population and one fixed sample. There is nothing random about this problem and the experiment is conducted once, so there is no "long run".

We pretend that the experiment was not conducted once, but an infinite number of times, that is, we consider all possible samples of the same size. We assume that each sample mean includes an "error", which is independently and normally distributed about zero. The sample mean now becomes our random variable, which we call our "statistic". We can now apply the t-test or z-test interpretation of probability.

We are now able to determine the probability of a randomly chosen sample mean having a value at least as extreme as our original sample mean. Note that we are implicitly assuming that the null hypothesis is true. This probability is our p-value which we apply to the original problem.

Remember that, in the t-tests for differences in means, there is a condition of equal population variances that must be examined. One way to test for possible differences in variances is to do an F test. However, the F test is very sensitive to violations of the normality condition; i.e., if populations appear not to be normal, then the F test will tend to reject too often the null of no differences in population variances.

You might like to use the following JavaScript to check your computations and to perform some statistical experiments for deeper understanding of these concepts:

Single Population t-Test

The purpose is to compare the sample mean with the given population mean. The aim is to judge the claimed mean value, based on a set of random observations of size n. A necessary condition for validity of the result is that the population distribution is normal, if the sample size n is small (say less than 30).

The task is to decide whether to accept a null hypothesis:

H0 = m = m0

or to reject the null hypothesis in favor of the alternative hypothesis:

Ha: m is significantly different from m0

The testing framework consists of computing a the t-statistics:

T = [( - m0) n1/2] / S

Where is the estimated mean and S2 is the estimated variance based on n random observations.

The above statistic is distributed as a t-distribution with parameter d.f. = n = (n-1). If the absolute value of the computed T-statistic is"too large" compared with the critical value of the t-table, then one rejects the claimed value for the population's mean.

This test could also be used for testing similar claims for other unimodal populations including those with discrete random variables, such as proportion, provided there are sufficient observations (say, over 30).

You might like to use Testing the Mean JavaScript in checking your computations. and Sample Size Determination JavaScript at the design stage of your statistical investigation in decision making with specific subjective requirements.

You might like also to use JavaScript Testing Two Populations.

Two Independent Populations

If an estimate is an unbiased such as sample mean, then it is a good idea to pool the estimates to get a single estimate from several relatively small samples. The pooled estimate is a “good” estimate when compared with each individual estimates.

Pooled Mean: Supposed we have m number of estimates (i), of sample size n(i), for the population expected value m, the pooled estimate is:

[S n(i)(i)] / [Sn(i)], both sums are over all values of i = 1, 2,. . ., m.

Pooled Variance: Since the sample variance is also unbiased estimate of population variance s2, therefore, it is a good idea to pool the estimates to get a single estimate from m number of estimates S(i)2, of sample size n(i), the pooled estimate is:

{} / {}, both sums are over all values of i = 1, 2,…, m.

We pool variance estimates for other good reasons. Depending on a particular reason, then the conclusion might have to be made explicitly conditional on e.g., the validity of the equal-variance model. There are several different good reasons for pooling:

You might like to use JavaScript Pooling the Means, and Variances.

Pooled Standard Deviation: Both the sample mean, and variance are unbiased estimates for the population parameters, m, and s2, respectively, however the sample standard deviation in NOT an unbiased estimate of population standard deviation s. This is so, because of an equality known as the Jensen's inequality when applied to a concave function, i.e., the square root of the unbiased variance estimate. Therefore, pooling standard deviation directly is meaningless; the best one can do to take the square root of the pooled variance

Notice that, when sample sizes are large and nearly equal, so that there is essentially no difference between the pooled and unpooled estimates of standard errors of paired-data samples, and degrees of freedom are nearly asymptotic. This rationale can fall apart for any other cases. One must pool variance rather than merely taking a shortcut in the computation of standard errors.

If you calculate the test without the assumption, you have to determine the degrees of freedom (d.f.). The formula works in such a way that d.f. will be less if the larger sample variance is in the group with the smaller number of observations. This is the case in which the two tests will differ considerably. A study of the formula for the d.f. is most enlightening, and one must understand the correspondence between the unfortunate design, having the most observations in the group with little variance, and the low d.f. and accompanying large t-value.

Applications: When doing t tests for differences in means of populations, for independent samples case:

  1. For differences in means that do not make any assumption about equality of population variances, use the standard error formula:

    [S21/n1 + S22/n2]˝,

    with d.f. = n = n1 or n2