Calculating Probabilities for Standard Normal Distribution | Applied Biostatistics | BIO733_Topic056

Duration: 6:37 851 transcript words EN summary

Video summary

The video explains that while calculating probabilities for a continuous random variable following a normal distribution requires integrating its probability density function, this process can be complex. To simplify these calculations, the standard approach is to convert any given random variable X into a standard normal random variable, often referred to as a Z-score. This transformation allows for the use of standardized tables containing cumulative probabilities, making it much easier to determine specific areas under the curve without performing difficult integrations manually. To find the probability corresponding to a specific Z-score using these tables, one must first visualize the standard normal curve and identify the area of interest. For instance, to find the cumulative area to the left of a positive Z-score like 1.15, the user locates the whole number and tenths digit in the column header (1.1) and then moves across to the hundredths digit row (0.05). The intersection provides the exact probability, such as 0.8749, which represents the total shaded area under the curve from negative infinity up to that Z-value. Similarly, for negative Z-scores like -0.24, the table is consulted by finding -0.2 in the column and 0.04 in the row, yielding a cumulative area of 0.4052. When the goal is to find an area other than the one directly provided by the cumulative table, specific strategies are employed based on the direction of interest. If the probability required is the area to the right of a Z-score, such as 1.23, the process involves first finding the area to the left (0.8907) and then subtracting this value from 1, since the total area under the normal curve is always equal to one. For cases involving the area between two specific Z-scores, like -0.75 and 1.23, the method involves calculating the cumulative area for both points individually and then subtracting the smaller cumulative area from the larger one to isolate the region of interest, resulting in a probability of 0.6641 for this example. In summary, the module emphasizes a systematic review of how to interpret standard normal tables to solve various probability problems efficiently. The core principles are that looking up a Z-score directly gives the area to its left; finding the area to the right requires subtracting that left-side value from one; and finding the area between two points requires subtracting the smaller cumulative probability from the larger one. By consistently sketching the curve, shading the relevant regions, and applying these arithmetic rules, students can accurately determine probabilities for any segment of the standard normal distribution without needing to perform complex integration every time.

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If X is a continuous random variable that follows the normal probability distribution and we have to find out the probability at certain X values. Then we always have to integrate the probability density function of the normal probability distribution to find out the probabilities. Which sometimes gets more complex. Therefore an easier way is to convert a random variable X into a standard normal random variable and use and this transformed function is much easily integrable as well as we have the standard normal table the tables for the standard normal probabilities available which we can use. So in this module we will learn how to use the the tables for the standard cumulative standard normal probabilities to find out the probabilities at certain Z score values. Over here this table is a table for the cumulative probabilities. And from now on as we know that for the continuous random variable the probabilities are always measured as area under the curve. So I will be using the area as a term to represent probabilities. So if we want to find the cumulative area that corresponds to a Z score of 1.15 then in this table of cumulative probabilities firstly we look at the column side. We will we where we will look at 1.1 and then we move across the row to the column under 0.05 which will give us the area to the left of Z equals to 1.15. And this area will be 0.8749, which is also shown in the normal curve. We're zero in the middle. So, this shaded area gives us the probability of Z being from any value that is less than equals to 1.15. In another case, as we know that a random variable X ranges from minus infinity to plus infinity, and the same is the case with the standard normal random variable, that it goes from minus infinity to plus infinity. So, Z can Z score can definitely take a value that is negative. So, if by any chance the Z gets the value minus 0.25 24, then firstly, we find minus 0.2 in the left-hand column. And then we move across the row to the column under 0.04. And hence the area to the left of Z will be minus Z equals to minus 0.24 is 0.4052. And again in the normal curve, it is represented by the shaded region. Whenever we need to find out the probability using Z scores and the table for the cumulative probabilities, I first suggest that sketch the standard normal curve. And shade the appropriate area under the curve for which we want the probability for. Secondly, find the area of the following direction for each case shown below. That if we are interested in finding out the area to the left of Z, then we find that area corresponds to Z in the standard normal table that has accumulative probabilities. And in the case if Z is equals to 1.23, then the area to the left will be 0.8907. In another situation, if our interest is to find out the area to the right of Z, then using the table for the standard normal cumulative probabilities, the area firstly, we find the area to the left of Z is equals to 1.23, which is 0.8907, and then we subtract it from 1. And as we know that the total area under the normal curve is 1, that's why we subtract the area to the left of 1.23 from 1, which is the total area, and this will help us to find out the area to the right of 1.23. Likewise, if we want to find out the area between two Z scores, let's say in this situation, we have 0 minus 0.75 and 1.23. And if we want to find out the area between these two points, at first, we will find the area to the left of 1.23, and we will also find the area to the left of minus 0.75. And then we will subtract the smaller area from the larger area, and hence we will get the area between these two points. And in this situation, if Z is 1.23 and the other Z value is minus 0.75, hence the area between these two points at their Z scores will be 0.6641. Just a quick review that if we want to find the area to the left of Z, we will directly look at the the table for the cumulative probabilities. And the value we will obtain from that table will always talk about the area to the left of Z score. And if we want to find out the area to the right of the Z score, we will firstly find the the area to the left of Z score by using the table of the proper cumulative probabilities and then subtract it from one. And lastly, if we need to find out the area between two Z scores, then we take the area for the those individual Z scores and then subtract the smaller value from the larger value.