Find P(z ≤ 0.42) Using The Standard Normal Table

In this article, we'll walk through how to find the probability that a standard normal random variable Z is less than or equal to 0.42, denoted as P(z ≤ 0.42). We'll make use of a standard normal table (also known as a Z-table) to find this probability. So, let's dive right in!

Understanding the Standard Normal Distribution

Before we get into the specifics, let's quickly recap what the standard normal distribution is all about. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It's a fundamental concept in statistics, and it's often used to model real-world phenomena. The beauty of the standard normal distribution lies in its simplicity and the fact that we can easily find probabilities associated with it using a standard normal table. This table gives us the cumulative probability, which is the probability that a standard normal random variable Z is less than or equal to a certain value z. Essentially, it tells us the area under the standard normal curve to the left of z. The standard normal distribution is symmetric around its mean (which is 0). This symmetry allows us to make certain inferences about probabilities. For example, P(Z ≤ 0) = 0.5, meaning there's a 50% chance that Z will be less than or equal to 0. This is because half of the area under the curve lies to the left of 0. When we look up values in the Z-table, we're finding the area under the standard normal curve from negative infinity up to the Z-value we're interested in. This area represents the cumulative probability P(Z ≤ z). So, by using the table, we can determine the likelihood of observing a value less than or equal to a specific point in our standard normal distribution. Remember, the total area under the curve is always equal to 1, representing 100% probability. This understanding forms the basis for hypothesis testing, confidence intervals, and many other statistical analyses.

Using the Standard Normal Table

The standard normal table, or Z-table, is a table that shows the area under the standard normal curve to the left of a given z-score. This area represents the cumulative probability P(Z ≤ z). To find P(z ≤ 0.42), we need to look up the value 0.42 in the Z-table. The Z-table typically has z-scores listed in the first column and the first row. The first column usually gives the z-score to one decimal place (e.g., 0.0, 0.1, 0.2, etc.), while the first row gives the second decimal place (e.g., 0.00, 0.01, 0.02, etc.). To find the probability corresponding to z = 0.42, we locate 0.4 in the first column and 0.02 in the first row. The value at the intersection of this row and column gives us P(z ≤ 0.42). Now, let's consider a slightly different scenario. Suppose we want to find P(z ≤ 1.65). We would locate 1.6 in the first column and 0.05 in the first row. The intersection of this row and column would give us the desired probability. Remember that the Z-table gives us the cumulative probability, i.e., the probability that Z is less than or equal to a specific value. If we want to find the probability that Z is greater than a value, we can use the fact that the total area under the curve is 1. For example, P(z > 1.0) = 1 - P(z ≤ 1.0). The Z-table is an indispensable tool in statistics, allowing us to easily find probabilities associated with the standard normal distribution. It's used extensively in hypothesis testing, confidence interval construction, and other statistical applications. By understanding how to use the Z-table, we can unlock a wealth of information about the standard normal distribution and its applications in various fields.

Finding P(z ≤ 0.42) From the Provided Table

Based on the small portion of the standard normal table you've provided, we don't have the exact value for z = 0.42. However, we can infer how to use the table. The given table shows:

From this, we understand how the table is structured. To find P(z ≤ 0.42), we would ideally look for the row corresponding to z = 0.42. Since that's not available, let's assume we had more data. Imagine the table also included these rows:

In this expanded table, we can directly find that P(z ≤ 0.42) = 0.6628. This means there's approximately a 66.28% chance that a standard normal random variable Z will be less than or equal to 0.42. Now, let's think about why this value makes sense. We know that P(z ≤ 0) = 0.5000, which is the probability that Z is less than or equal to the mean (0). Since 0.42 is greater than 0, we expect the probability P(z ≤ 0.42) to be greater than 0.5000. Our value of 0.6628 confirms this. The closer z is to 0, the closer P(z ≤ z) will be to 0.5000. As z increases, P(z ≤ z) also increases, approaching 1 as z goes to infinity. This is because the cumulative probability represents the area under the curve to the left of z, and as z moves further to the right, it encompasses more and more of the area. So, by using the standard normal table, we can easily find the probabilities associated with different values of z, giving us valuable insights into the behavior of the standard normal distribution. Remember to always double-check the table structure and ensure you're reading the correct row and column to obtain the accurate probability.

Approximation and Interpolation (If Necessary)

In many real-world scenarios, you might not find the exact z-value you're looking for in the standard normal table. In such cases, you can use approximation or interpolation to estimate the probability. Approximation involves selecting the closest available z-value in the table and using its corresponding probability as an estimate. For instance, if you want to find P(z ≤ 0.423) and the table only has values for 0.42 and 0.43, you could approximate P(z ≤ 0.423) by using either P(z ≤ 0.42) or P(z ≤ 0.43), depending on which is closer. A more accurate method is interpolation. Interpolation involves estimating the probability based on the probabilities of the two nearest z-values in the table. Linear interpolation is a common technique that assumes the probability changes linearly between the two known points. To illustrate, let's say you want to find P(z ≤ 0.423), and you have the following values from the table: P(z ≤ 0.42) = 0.6628 and P(z ≤ 0.43) = 0.6664. Using linear interpolation, you can estimate P(z ≤ 0.423) as follows: P(z ≤ 0.423) ≈ P(z ≤ 0.42) + 0.3 * (P(z ≤ 0.43) - P(z ≤ 0.42)) = 0.6628 + 0.3 * (0.6664 - 0.6628) = 0.6628 + 0.3 * 0.0036 = 0.6628 + 0.00108 = 0.66388. So, P(z ≤ 0.423) ≈ 0.66388. Interpolation provides a more precise estimate than simple approximation, especially when the z-value falls midway between two values in the table. Keep in mind that interpolation assumes a linear relationship between the z-value and the probability, which may not always be perfectly accurate. However, for most practical purposes, linear interpolation provides a sufficiently accurate estimate. When using approximation or interpolation, it's crucial to be aware of the potential error introduced by these methods. The accuracy of the estimate depends on the spacing of the z-values in the table and the degree of non-linearity in the relationship between z and the probability. In general, the closer the z-values are in the table, the more accurate the approximation or interpolation will be.

Conclusion

So, while the provided table snippet was limited, we've shown how to find P(z ≤ 0.42) using a standard normal table. If we had a more complete table, we would simply look up the value corresponding to z = 0.42. Based on our example expanded table, P(z ≤ 0.42) = 0.6628. Remember, the standard normal table is a powerful tool for finding probabilities associated with the standard normal distribution, and understanding how to use it is essential for various statistical applications. Keep practicing, and you'll become a pro at using the Z-table in no time! And that's a wrap, folks! Hope this helped you understand how to find probabilities using the standard normal table. If you have any questions, feel free to ask. Happy calculating!