Standard normal distribution areas

Appendix 1 Standard normal distribution areas Appendix 2 Percentiles of t distributions Appendix 3 Percentiles of distributions Appendix 4 Percentiles of F distributions (a = 0.05) Appendix 5 Values of q for Tukey s HSD test (a = 0.05)... 

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

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).

Because (a) is the area in the lower tail of the normal distribution, Za is called the ath quantile of the standard normal distribution, (or the (100)(a)th percentile). A useful identity follows directly from the symmetry of the Gaussian distribution in Equation A-2 (4). 

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... 

Enter table by Z-value to obtain cumulative area entry. As an example, area for Z = 1.96 (entry at row 1.9, column 0.06) is 0.9750, indicating that 97.5% of the standard normal distribution is below this Z-value, and 2.5% is above. Areas for negative Z-values are calculated by subtracting the area for the positive Z-value from 1. For example, the area for Z = -1.96 is calculated as 1 - 0.9750 or 0.0250. 

With respect to equation (3.17), the standard normal distribution function has a mean value fix = 0 and a standard deviation cr = 1. The area under a standard normal distribution is equal to unity. The standard normal distribution function can be written as N(0,1). 

The random number generated is the cumulative probability, and the cumulative probability is the area under the standard normal distribution curve. Since the standard normal distribution curve is symmetrical, the negative values of Z and the corresponding area are found by symmetry. For example, as described in the two previous problems,... 

Table A.1 contains values of the right-tail areas of the standard normal distribution, from 2 = 0.00 to 3.99. The first column contains the value of 2 to the first decimal place, while the top line in the table gives the second decimal. To find the value of the tail area for a given value of 2 we look in the table at the appropriate intersection of line and column. The value corresponding to 2 = 1.96, for example, is at the intersection of the line corresponding to 2 = 1.9 and the column headed by 0.06. This value, 0.0250, is the fraction of the total area located to the right of 2 = 1.96. Since the curve is symmetrical about the mean, an identical area is located to the left of 2 = —1.96, in the other half of the Gaussian (Fig. 2.4). The sum of these two tail areas, right and left, equals 5% of the total area. From this we can conclude that the remaining 95% of the area lies between 2 = —1.96 and 1.96. If we randomly extract a value of 2, there is 1 chance in 20 (5%) that this value will lie below —1.96 or above 1.96. The other 19 times, chances are that it will fall inside the [—1.96, 1.96] interval.

Fig. 2.4. Symmetric interval about the mean, containing 95% of the total area under the standard normal distribution curve.

Table A.2 contains t values for some right-tail areas of Student s distribution. The areas appear at the top of the table, in boldface. Like the standard normal distribution, the t distribution is symmetric about a zero mean, so we only need to know the tail values for one side.

Fig. 3.8. Random sample of 10 elements taken from a standardized normal distribution. Each element represents a region whose area is equal to 1/10 of the total area under the curve.

The standard normal distribution results from the special case wherein pi = 0 and area under the curve from —< to + > is exactly 1.0. If one can develop a table of random numbers for a uniform distribution over the interval 0-1, it is possible to map a set of equivalent values for the standard normal distribution, as in Figure 10. The value along the ordinate represents the probability that the random variable X lies in the interval —< to x. For any random number we can compute the equivalent value x. This latter value is called the random normal deviate. 

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. 

ACLE II Areas Under the Standard Normal Curve-the Values were Generated using the Standard Normal Distribution Function of Excel (coi inued) . t. ... 

In Table 19.H> z = 1 represents one standard deviation above the mean and 34.13% of the total area under a standard normal curve. On the other hand, z = — 1 represents one stan dard deviation below the mean and 34.13% of the total area, as shown in Figure 19.7. Therefore, for a standard normal distribution, 68% of the data foil in the interval of z = — 1 to z = 1 (—s to s). Similarly, z = —2 and z = 2 (two standatd deviations helow and above the... 

From Fig. 5.2 or a table of normal distribution areas and ordinates [2,3,5], the probability of obtaining such a value of t is P ( t > 2.12) = 0.5000 — 0.48300 = 0.017. This probability value 0.017 is highly imlikely because it lies outside of two standard deviations from the mean thus we reject the null hypothesis Hq that /x 15.5 for our second experiment, and it is concluded that the new chemical process produces a new value for the concentration of A which might be detrimental to the manufacturing process. 

Table 14.1 Areas of the standard normal distribution between -Z and +Z.

The normal probability distribution may be obtained from the standard normal frequeney distribution (Fig. 14.2). The probability that a reading will fall between (t<7)i and ta)2 in Fig. 14.4 will be the area under the frequency curve between these two points. Table 14.3 gives the probability for a value to fall between x = 0 and x = ta on the standard normal distribution eurve. This corresponds to the area under the curve between these two points. 

Figure 14.4. The standard normal distribution curve showing a cross-hatched area below the curve between ta)i and (10)2. The area of this cross-hatched region is the probability (P) that a point will fall within this area.

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