Standard Normal Probability

In order to compensate for the uncertainty incurred by taking small samples of size n, the / probability distribution shown in Figure 3.2 is used in the calculation of confidence intervals, replacing the normal probability distribution based on z values shown in Figure 3.1. When n > 30, the /-distribution approaches the standard normal probability distribution. For small samples of size n, the confidence interval of the mean is inflated and can be estimated using Equation 3.9... 

Figure 1 Standardized normal probability curve and characteristic parameters, the mean and standard deviation...

Find the mean value and the root-mean-square value of cos (x) for —00 < X < 00, assuming the standard normal probability distribution... 

Figure 7.5 (a) Standardized normal probability density function. 

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the cui ve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical beU-shaped cui ve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

Using the SND theory from Appendix I, the probability of failure, P, ean be determined from the Standard Normal variate, z, by ... 

We already know SM = 2.34 because it is the positive value of the Standard Normal variate, z, calculated above. The probability of failure per application of load... 

From the Standard Normal Distribution (SND) it is possible to determine the probability of negative elearanee, P. 

The area under the eurve to the left of i, — /i, = 0 relates to the probability of negative elearanee. This area ean be found from the SND table (Table 1, Appendix I) by determining the Standard Normal variate, z, where ... 

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

TABLE 20.5.2 Standard Normal, Cumulative Probability in Rigbt-Hand Tail (for Negative Values of z, Areas Are Found by Symmetry) ... 

Table 20.5.2 also can be used to determine probabilities concerning normal random variables tliat are not standard normal variables. The required probability is first converted to tm equivalent probability about a standard normal variable. For example if T, the time to failure, is normally distributed with mean p = 100 and stanchird deviation a = 2 tlien (T - 100)/2 is a standard normal variable and... 

The probability function for the standard normal distribution is then... 

For np > 5 and n( 1 - p) > 5, an approximation of binomial probabilities is given by the standard normal distribution where z is a standard normal deviate and... 

The normal probability distribution function can be obtained in Microsoft Excel by using the NORMDIST function and supplying the desired mean and standard deviation. The cumulative value can also be determined. In MATLAB, the corresponding command is randn. 

The uniformity of the particles as measured by the standard deviation for a normal probability curve was found to be a function of flow rate, nozzle size (better for smaller nozzles), nozzle length (decrease in nozzle length decreases uniformity), etc. A decrease in interfacial tension is insufficient to cause change in uniformity. 

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. 

Figure 1.8 explains graphically how probabilities and quantiles are defined for a normal distribution. For instance the 1 %-percentile (p = 0.01) of the standard normal distribution is —2.326, and the 99%-percentile (p 0.99) is 2.326 both together define a 98% interval. 

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

The probability that the variable x takes a value between a and b is given by the area under the graph of the probability distribution between x=a and x=b. This is illustrated in Figure 21.3, where the shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. the probability P(Z>z). The total area under the graph is equal to 1, and because of the symmetry of the normal distribution it follows that the area of any one half is equal to 0.5. For any normal... 

Figure 21.3 The standard normal distribution curve. The shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. P(Z>z).

Figure 2.1. The standardized normal or Gaussian distribution. The shaded area as a fraction as the entire area under the curve is the probability of a result between Xj and X2.

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

Using Table 1.3 i.e. Table B for standard normal distribution, determine probabilities that correspond to the following Z intervals. 

The chi-square distribution was discussed briefly in the earlier section on probability distributions. Suppose we have (k+1) independent standard normal variables. We then define % as the sum of the squares of these (k+1) variables. It can be shown that the probability density function of % is ... 

According to the important theorem known as the central limit theorem, if N samples of size n are obtained from a population with mean, fi, and standard deviation, a, the probability distribution for the means will approach the normal probability distribution as N becomes large even if the underlying distribution is nonnormal. For example, as more samples are selected from a bin of pharmaceutical granules, the distribution of N means, x, will tend toward a normal distribution with mean /j and standard deviation <7- = a/s/n, regardless of the underlying distribution. 

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