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1. Calculating Normal Probabilities
An important skill in hypothesis testing is being able to identify probabilities from a distribution. In this lesson, we’ll learn how to calculate probabilities using a normal distribution, which is used in many hypothesis tests.
Lesson Goal (00:19)
The goal of this lesson is to calculate probabilities using the normal distribution.
Overview of the Problem (00:26)
The problem in this lesson relates to scores on an exam taken by many students, where the population mean is 240 and the standard deviation is 50.. Our aim is to calculate the probability of various possible values from this distribution.
Calculating a Probability from the Distribution (00:39)
The first problem requires us to calculate the probability of observing a value above 300. This is equivalent to the area to the right of 300 under the normal distribution. The area to the left of 300 represents the probability of observing a value below 300.
To find this area, we first need to calculate the Z-Score for this observation, to transform it to a point on the standard normal distribution. Next, we use a Z-table to identify the area below this Z-Score. The table we use for this purpose can be found here. We use the rows and columns to identify the Z-Score of interest, then find the corresponding probability in the body of the table. The table gives us the probability of observing a value below 300, and we can subtract this probability from one to find the probability of a score above 300.
Note that we can also use statistical software to find these probabilities instead of a Z-table, but we use the table in this lesson so that you fully understand how the process works.
Calculating the Probability Within a Range (02:39)
We can find the probability between two values by finding the probabilities to the left of the individual values, then finding the difference between these probabilities. For example, to find the probability of observing a value between 200 and 280, we find the probability of observing a value less than 200, and the probability of observing a value less than 280 using the same technique as before. We then find the difference between these probabilities, which represents the probability of observing a value between 200 and 280.
Finding the Value for a Probability (06:05)
Using the normal distribution, we can also find the value from the distribution which has a certain probability above and below it. For example, we aim to find the test score with a probability of 0.9 of being below it.
To do this, we find the appropriate Z-Score by locating the value of 0.9 in the Z-table, and reading the corresponding Z-Score. We then use the Z-Score formula to identify the value from the normal distribution that corresponds to this Z-Score.