Standard normal table

Since we have only one observation for X, the test statistic is simply its standardized value, Z = (X — p) / ax. The standard normal tables give the distribution of this statistic. 

The value of zq from the standard normal table (see Problem NUC.9 in Chapter 24) is then 2.05, which corresponds to the number of standard deviations above the mean tank concentration. According to the central limit theorem, the standard deviation of the mean is given by... 

From standard normal table (see Problem NUC.9, Chapter 24). 

First determine the values of the standard normal variable, Z, for component A using the 10 random numbers given in the problem statement and a standard normal table. Then calculate the lifetime of thermometer component A, using the equation for T. 

H. Use of Standard Normal Table with Rate Equations (09). 271... 

Using a standard normal table, the probability corresponding to the area is given by j8 = 0.0217. Note however, that the /3 error should not include that area to the left of x = 95.1. The probability that X will be less than this value given that fx = 110 is determined by P Z < Zf), where... 

This means that the posterior density of /z is j/ plus times a Student s t with n — 1 degrees of freedom. This means we can do the inferences for treating the unknown variance as if it had the value d" but using the Student s t table instead of the standard normal table. 

Table 1 Area under the cumulative Standard Normal Distribution (SND)...

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... 

From Table 20.6.2, z, is found to be -1.28. The corresponding simulated value of Ta is obtained by noting tliat (Ta - 100)/20 is a standard normal variable. Therefore, tlie simulated value of Ta corresponding to z, = -1.28 is... 

Since (In Z - p)/a is a standard normal variable, we can refer to a table tliat gives areas under tlie standard nonnal curve (see Table 21.5.2) to find tliat... 

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... 

Using a standard normal distribution table, Z = Zoos = 1.96 Zp = Zo.i = 1.65. Substituting into the equation, we have... 

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. 

Matter Volatile and Loss on Ignition. The content of matter volatile in a pigment is determined by drying a sample in an oven at 105 2°C. This normally gives a measure of the moisture content. For standards, see Table 1 ( Matter volatile ). 

In this case, pz — 0 standard normal distribution is illustrated in Fig. 1.4, and a brief tabulation given in Table 1.3 from Table B. 

Table 1.3 Abbreviated table of standard normal distribution...

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

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... 

A few values of the t-distribution are given in an accompanying table. We note that t values are considerably higher than corresponding standard normal values for small sample size but as n increases, the t-distribution asymptotically approaches the standard normal distribution. Even at a sample size as small as 30, the deviation from normality is small, so that it is possible to use the standard normal distribution for sample sizes larger than 30 (n>30) and in most cases, for n<30 t-distribution is used. This is equivalent to assuming that Sx is an exact estimate of ox at large sample sizes (n>30). 

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