How to Use the Normal Distribution to Solve Common Core Algebra 2 Homework Problems
The normal distribution is a type of probability distribution that describes how data values are spread around a mean (average) value. It is also known as the bell curve or the Gaussian distribution because of its shape. The normal distribution has many applications in statistics, science, engineering, and other fields.
In this article, we will show you how to use the normal distribution to solve some common core algebra 2 homework problems. We will also provide you with some online resources where you can find more solutions and explanations for algebra 2 common core textbooks.
Example 1: Finding the Probability of a Data Value
One of the most common uses of the normal distribution is to find the probability of a data value falling within a certain range or interval. For example, suppose you want to find the probability that a randomly selected student from a class has a height between 64 inches and 68 inches, given that the mean height of the class is 66 inches and the standard deviation is 3 inches.
To solve this problem, we need to use the following formula:
P(a < X < b) = P(Z < (b - Î¼) / Ï) - P(Z < (a - Î¼) / Ï)
where X is the random variable of interest (height in this case), Î¼ is the mean, Ï is the standard deviation, Z is a standard normal random variable (with mean 0 and standard deviation 1), and P is the probability function.
Therefore, the probability that a randomly selected student from the class has a height between 64 inches and 68 inches is about 0.4972 or 49.72%.
Example 2: Finding the Data Value Corresponding to a Given Probability
Another common use of the normal distribution is to find the data value that corresponds to a given probability or percentile. For example, suppose you want to find the height that separates the top 10% of students from the rest of the class, given that the mean height of the class is 66 inches and the standard deviation is 3 inches.
To solve this problem, we need to use the inverse of the formula we used in example 1:
X = Î¼ + ZÏ
where X is the data value we are looking for, Î¼ is the mean, Ï is the standard deviation, and Z is the standard normal random variable that corresponds to the given probability or percentile.
To find Z, we need to use a table or a calculator that gives us the values of Z for different probabilities or percentiles. For example, if we want to find Z for the top 10%, we need to look for the value of Z that gives us a probability of 0.9 or a percentile of 90%. Using a table or a calculator, we can find that Z = 1.28.
Using this value of Z, we can calculate:
X = Î¼ + ZÏ
= 66 + (1.28)(3)
Therefore, the height that separates the top 10% of students from the rest of the class is about 69.84 inches.
Online Resources for Algebra 2 Common Core Homework Answers
If you need more help with your algebra 2 common core homework problems, you can check out some online resources that provide solutions and explanations for various textbooks. Here are some examples: