Examples of zvalue in the following topics:



 A normal probability table, which lists Z scores and corresponding percentiles, can be used to identify a percentile based on the Z score (and vice versa).
 Generally, we round Z to two decimals, identify the proper row in the normal probability table up through the ﬁrst decimal, and then determine the column representing the second decimal value.
 We can also ﬁnd the Z score associated with a percentile.
 For example, to identify Z for the 80th percentile, we look for the value closest to 0.8000 in the middle portion of the table: 0.7995.
 We determine the Z score for the 80th percentile by combining the row and column Z values: 0.84.

 The first column titled "Z" contains values of the standard normal distribution; the second column contains the area below Z.
 For example, a Z of 2.5 represents a value 2.5 standard deviations below the mean.
 A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:
 where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution.
 If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard deviation of 1.

 If it is 1.5 standard deviations below the mean, then its Z score is 1.5.
 Using µSAT = 1500, σSAT = 300, and xAnn = 1800, we ﬁnd Ann's Z score:
 Observations above the mean always have positive Z scores while those below the mean have negative Z scores.
 SAT score of 1500), then the Z score is 0.
 One observation x1 is said to be more unusual than another observation x2 if the absolute value of its Z score is larger than the absolute value of the other observation's Z score: Z1 > Z2.


 That is, Z 1, Z 2, Z 3, and Z 4 must be combined somehow to help determine if they – as a group – tend to be unusually far from zero.
 A first thought might be to take the absolute value of these four standardized differences and add them up:
 Z 1  + Z 2  + Z 3  + Z 4  = 4.58
 The test statistic X 2 , which is the sum of the Z 2 values, is generally used for these reasons.
 Using this distribution, we will be able to obtain a pvalue to evaluate the hypotheses.

 We may also generalize the confidence level by using a placeholder z* .
 In this section, we provide the computed standard error for each example and exercise without detailing where the values came from.

 The simplest way to ﬁnd the shaded area under the curve makes use of the Z score of the cut oﬀ value.
 With µ = 1500, σ = 300, and the cutoﬀ value x = 1630, the Z score is computed as
 However, the percentile describes those who had a Z score lower than 0.43.
 To ﬁnd the area above Z = 0.43, we compute one minus the area of the lower tail:
 The picture shows the mean and the values at 2 standard deviations above and below the mean.

 The computation of a confidence interval on the population value of Pearson's correlation (ρ) is complicated by the fact that the sampling distribution of r is not normally distributed.
 The conversion of r to z' can be done using a calculator.
 This calculator shows that the z' associated with an r of 0.654 is 0.78.
 The Z for a 95% confidence interval (Z.95) is 1.96, as can be found using the normal distribution calculator (setting the shaded area to .95 and clicking on the "Between" button).
 The r associated with a z' of 1.13 is 0.81 and the r associated with a z' of 0.43 is 0.40.