Final answer:
The probability that z = 0 in a standard normal distribution is technically 0, as it represents an exact value on a continuous distribution. Z-scores indicate the number of standard deviations a value is from the mean, with z = 0 being the mean itself.
Step-by-step explanation:
The question 'what is the probability that z = 0?' refers to the standard normal distribution where the mean (μ) is zero and the standard deviation (σ) is one. In a standard normal distribution, a z-score represents the number of standard deviations a value x is from the mean. The probability that z = 0 is the probability that a value lies exactly at the mean of the distribution, which for a continuous distribution is technically 0. However, we typically are interested in the area under the curve surrounding z=0 to find probabilities for a range of values.
For example, if z represents the probability that a child takes more than two minutes to eat a donut given that they have been eating for more than 1.5 minutes, there isn't a single value for z unless it is defined in the context of the question. Probability distributions are used to calculate probabilities for ranges or intervals, not exact instances.