Statistics standard normal distribution

The t (Student s t) distribution is an unbounded distribution where the mean is zero and the variance is v/(v - 2), v being the scale parameter (also called degrees of freedom ). As v -> < , the variance —> 1 (standard normal distribution). A t table such as Table 1-19 is used to find values of the t statistic where... 

A conceptual definition of the follows from consideration of a set of numbers drawn at random from the standard normal distribution, the one with mean zero and standard deviation one. Ordering this set of numbers gives a sequence called order statistics. The are the... 

The normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

The number 0.128 is the largest acceptable true CV p for which the net error would not exceed +25% at the 95% confidence level. The number 1.96 is the appropiate Z-statistic (from tables of the standard normal distribution) at the same confidence level. 

To determine the critical region, we must know the distribution of the test statistic. In this case, Z is distributed as the standard normal distribution. With H0 [t<48 and a=0.05, we determine that the critical region will include 5% of the area on the high end of the standard normal curve Fig. 1.6. The Z-value that cuts off 5% of the curve is found to be 1.645, from a table of... 

Use standard normal distribution (z-statistic) for large populations (number of samples... 

Similarly to the case of the standard normal distribution, the critical values can be obtained from a series of tabulated values or from statistical software. A number of percentiles of various t distributions are provided in Appendix 2. It is important to note that there is not just one t distribution there are many of them, and their shapes are determined by the number of degrees of freedom. As either low or high values of the test statistic could lead to rejection, the hypothesis test is considered a two-sided test. The probability of committing a type I error is O.OS, but, because the critical region is evenly split between low values and high values, the probability of committing a type I error in favor of one direction (for example, large values of t) is a/2. 

Under the null hypothesis of equal population means, the test statistic follows a t distribution with Kj 4- 2 - 2 degrees of freedom (df), assuming that the sample size in each group is large (that is, > 30) or the underlying distribution is at least mound shaped and somewhat symmetric. As the sample size in each group approaches 200, the shape of the t distribution becomes more like a standard normal distribution. Values of the test statistic that ate fat away from zero would contradict the null hypothesis and lead to its rejection. In particular, for a two-sided test of size a, the critical region (that is, those values of the test statistic that would lead to rejection of the null hypothesis) is defined by t[Pg.148]

As -7.98 < -1.968, the null hypothesis is rejected in favor of the alternate one. The mean change from baseline for the test treatment group is significantly different from the placebo group s at the a = 0.05 level. To determine the p value associated with this test, we need statistical software or an extensive look-up table. Given the large sample size in this example, we can use the percentiles of the standard normal distribution to approximate the p value. These study results allow us to conclude that the test treatment is... 

The derivation of these two parameters is beyond the scope of this text. Applying a familiar mathematical operation (standardization of a normally distributed random variable), we obtain an alternate test statistic, which has an approximate standard normal distribution ... 

Values of this test statistic can then be compared with the more familiar critical values of the standard normal distribution. 

Under the null hypothesis, this test statistic follows a standard normal distribution. The null hypothesis is rejected because the test statistic falls in the rejection region for a two-sided test of a = 0.05 based on the standard normal distribution (Z < - 1.96 ot Z > 1.96). 

According to the central limit theorem, the probability density distribution of x for a sample of size n will be given by a standard normal distribution with the test statistic defined [2] as... 

When the population variance is not known, the confidence interval for a population mean cannot be determined using a standard normal distribution, but it can be determined with the t distribution. As discussed in Section 3.2.5, one defines the test statistic t in terms of x, p, n, and the sample variance (see Equation (3.13)). By replacing the generic test statistic y in Equation (3.17) with t, the interval for p is defined [2] by... 

This interval is larger when dehned by a t statistic. Some difference is due to the sample variance being slightly larger than the population variance in the previous example, but the major difference is due to the t distribution being much wider than the standard normal distribution when the sample size is small. The following example demonstrates the effect of sample size on conhdence intervals. 

The number of bootstrap estimates of the statistic less than the observed test statistic is calculated and called p. Set b = 4>-1(p/B) where 4> 1( ) is the inverse of the standard normal distribution. 

A number of statistical techniques exist for assessing whether a given distribution departs from normality. Historically, the most commonly used method in crystallography is the normal probability plot [29, 30]. In this procedure, the n observations in a sample are normalised to zero mean and unit variance, and then ranked. The resulting ranked quantities are plotted against the order statistics expected for a random sample of size n taken from the standard normal distribution. Departures from a straight-line plot indicate non-normality. Various type of departure (e.g. bowed lines or S-shaped curves) are characteristic of particular sorts of deviations from normality. 

In statistical analysis involving normal distributions some other types of distributions are encountered frequently. The t-distribution is encountered e.g. in the calculation of confidence intervals in various situations. Its limiting distribution is the standard-normal distribution. The ( -distribution Is the sum of squares of several standard-normal distributed variables. It may be encountered in tests on normality of data. 

Both fnnctions are tabulated in mathematical handbooks (Ref 1). The function P gives the goodness of fit. Call %o the value of at the minimum. Then P > 0.1 represents a believable fit i( Q > 0.001, it might be an acceptable fit smaller values of Q indicate the model may be in error (or the Q are really larger.) A typical valne of x for a moderately good fit is x V Asymptotically for large v, the statistic becomes normally distributed with a mean v anaa standard deviation V( (Ref 231). 

Proceeding as in the example with 10 points, we can draw a normal plot of the 15 values and use it to test hypotheses about whether the effects exist or not. The data needed for this are listed in Table 3.9, where each effect of Table 3.8 is associated with a cumulative probability value. The easiest way to draw the normal plot of these values is to use one of the many statistical computer programs available. If you do not have access to one of these programs, you can use a standard linear scale, putting the values of the effects on the abscissa, but marking the values corresponding to the standardized normal distribution (z, in the last column of the table) for the center point of each interval on the ordinate axis. 

Confidence intervals are somewhat arbitrary but the generally accepted criterion requires a 95% probability of the true value falling within the confidence interval. For very large samples, (n > 30), the confidence interval is calculated assuming the distribution is Gaussian. For sample sizes less than 30, the value of s fluctuates substantially from sample to sample and thus the distribution is no longer a standard normal distribution. For this case, we represent the distribution with a statistic that is known as the Student s r-statistic. 

In statistics, it is customary and easier to normalize the mean and the standard deviation values of an experiment and work with what is called the standard normal distribution, which has a mean value of zero (jc = 0) and a standard deviation value of 1 (t = 1). To do this, we define what commonly is referred to as a tr score according to... 

In Equation (19.9), z represents the number of standard deviations from the mean. The mathematical fimction that describes a normal-distribution curve or a standard normal curve is rather complicated and may be beyond the level of your current understanding. Most of you will learn about k later in your statistics or engineering classes. For now, using Excel, we have generated a table that shows the areas under portions of the standard normal-distribution curve, shown in Table 19.11. At this stage of your education, it is important for you to know how to use the table and solve some problems. A more detailed explanation will be provided in your future classes. We will next demonstrate how to use Table 19.11, usii a number of example problems. 

The integrals (E.2.45) thus (E.2.44) giving the probability can be computed by recursion formulae they can be found in statistical tables. The information is not restricted to the standard normal distribution. Let X be a general Gaussian random variable. Let us introduce the random variable... 

An objective criterion has been developed to distinguish nonstatlonarlty from statlonarlty when a single finite realization of a stochastic process Is given. The criterion consists of several transformations leading to a set of standard normal distributed random variables as an equivalent notion of statlonarlty. Statlonarlty can therefore be tested by a statistical test. Correlations between estimates, important for investigating narrow banded processes, can be taken into account. 

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