Summary
Highlights
A normal distribution is a symmetrical curve about the mean, with the total area under the curve representing a probability of 1. The standard normal distribution is a specific type where the mean is 0 and the standard deviation is 1. This standardization is crucial for using z-scores, which measure how many standard deviations a value is from the mean.
The formula for a z-score is (value - mean) / standard deviation. The video demonstrates this with an example of commuting time. For instance, to find the probability of a commute less than 40 minutes (with mean 44 and standard deviation 3.5), convert 40 to a z-score of -1.14. Using a z-score table, this corresponds to an area (probability) of 0.1271, meaning a 12.71% chance.
Z-score tables provide the area to the left of a given z-score. Both negative and positive z-score tables exist, and they always show the probability of a value being less than or equal to the z-score. It's important to note that for continuous variables, the probability of an exact value is zero.
To find the probability of a value falling between two points (e.g., between 38 and 45 minutes commute), first convert both values to z-scores (-1.71 and 0.29). Look up their corresponding probabilities in the z-table (0.0436 and 0.6141). Subtract the smaller probability from the larger one (0.6141 - 0.0436 = 0.5705) to get the probability between the two values (57.05%).
To find the probability of a value being greater than a specific point (e.g., longer than 45 minutes commute), first find its z-score (0.29) and the area to its left from the table (0.6141). Since the total area under the curve is 1, subtract the area to the left from 1 (1 - 0.6141 = 0.3859) to get the area (probability) to the right (38.59%).