## Z-Test

### Z-score history

• Edward Altman, NYU (New York University)Stern Finance Professor , developed and introduced the Z-score formula in the late 1960s providing investors an idea about overall financial health of a company.
• Over the years, Altman has continued to revaluate his Z-score over the years.
• Z-Score is frequently used in testing credit strength.
• In general, a Z-score below 1.8 suggests a company might be headed for bankruptcy, while a score closer to 3 suggests a company is in solid financial positioning.

A z-test is a statistical test to determine whether two population means are different when the variances are known and the sample size is large.

• The test statistic is assumed to have a normal distribution.
• Altman Z-score or Z-scores are used in statistics to measure an observation’s deviation from the group’s mean value.
• Z-score= 0, it indicates that the data point’s value is identical to mean.
• Z-score= 1.0 would indicate a value that is one standard deviation from mean.
• Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

### Z-test Vs t-Test

• Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size.
• Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known.
• If the standard deviation of the population is unknown, the assumption of the sample variance equaling the population variance is made.
• The z-test is best used for greater-than-30 samples because, under the central limit theorem, as the number of samples gets larger, the samples are considered to be approximately normally distributed.

### The Difference Between Z-Scores and Standard Deviation

• To Find The Area to The Right of a Positive Z Score-considering that the total area under the bell curve is always 1 (which is equivalent to say that is 100%), for 95% confidence level, we will need to subtract the area(0.975) from 1 (1-0.975= 0.025) as 0.975 was the area to the left (shown in green in top fig)of z = 1.96.
• α= 1-CL, (significance level (α) for 95% confidence level will be 1-0.95 = .05)
• 1-α /2 is the value from the standard normal distribution holding 1- α/2 below it. For example, if α=0.05, then 1- α/2 = 0.975 and Z=1.960 on positive Z score table.
• α /2 is the value from the standard normal distribution holding α/2 below it. For example, if α=0.05, then α/2 = 0.025 and Z=1.960 on negative Z-score table.
• 1-α /2 positive Z score table.
• α /2 negative Z-score table.
• Let’s say, we want to find the p(Z < 2.13). From Z table probability value is 0.9834. Therefore p(Z < 2.13) = 0.9834. and p(Z > 2.13) = 1 – 0.9834 = 0.0166.
• To Find The Area To The Left of a Negative Z Score-we just need to disregard the negative sign and then simply subtract the area read from Z table from 1.
• CL=95%, Z score=1.96, entire Area under curve from left up to the positive Z value of 1.96 will be 0.975 (can be calculated from CI or read from Z table).
• Area to the left of positive Z score value of 1.96= AL=(1+0.95)/2 =0.975