Z-scores abd normal distribution

Question

Z-scores are used to interpret scores from any normal distribution or ratio scores. I was asked if the Z=0. Do you knw the answer?

Answer 1

The answer is that it depends on how the question is interpreted.  I can think of 2 ways to interpret it:

 

1. You're doing a 2-tailed hypothesis test to determine if Z is zero, with some level of confidence (that has not been specified in the question).  For example, if you wanted to test at a 95% confidence level, then the critical Z value is 1.96.  That value can be found in a standard normal table, or an online calculator, such as http://www.measuringu.com/pcalcz.php  So, if you ran an experiment and computed the mean and standard deviation for your data sample, then

 

mean = (sum of X_i)/n, i = 1, ...n in the sample data

standard deviation = square root[ (sum of (X_i - mean)2)/(n - 1) ]

 

n = sample size

 

X = random observation which is distributed as Normal(mean, standard deviation)

Z = (X - mean)/(standard deviation) follows the normal distribution with mean 0 and standard deviation 1, or
Z = normalized observation which is distributed as Normal(0, 1)

 

See https://statistics.laerd.com/statistical-guides/measures-of-spread-standard-deviation.php for more details.

 

H₀: Z = 0    (null hypothesis)

H₁: Z <> 0  (alternate hypothesis)

 

See http://statistics.about.com/od/Inferential-Statistics/a/The-Difference-Between-The-Null-Hypothesis-And-Alternative-Hypothesis.htm for more details.

 

If you computed a Z-score that was less than -1.96 or greater than 1.96, you would say that the null hypothesis of Z=0 was rejected at the 95% confidence level.

 

2. Another possible, but less likely, way that the question could be interpreted is what is the probability that the Z-score is exactly 0, or P(Z = 0)?  That is a bit of a trick question and the answer is 0 because the probability of a specific score with no confidence level specified is the area under the normal curve at that single point.