6.1.4 The Standard Normal Distribution - Given a range of z scores, find areas or probabilities.

This paragraph introduces the concept of finding areas or probabilities associated with z-scores in the context of continuous probability distributions, specifically the standard normal distribution. It emphasizes that the total area under the curve of a probability density function is 1, and that areas to the left of a z-score represent probabilities. The paragraph also mentions two methods for finding these probabilities: using technology like Excel or a z-table (table A2), with a preference for technology due to its accuracy. The author explains that table A2 is important for exams and should be understood for non-standard normal distributions, which can be converted to standard form using specific formulas.

The second paragraph delves into the specifics of using table A2 to find probabilities associated with z-scores. It clarifies that table A2 is designed for the standard normal distribution and explains the layout of the table, which includes separate pages for negative and positive z-scores. The paragraph describes how the body of the table represents the area to the left of a given z-score and how to infer the area to the right or between two z-scores. It also discusses the importance of not confusing z-scores with areas and the process of looking up z-scores on the outside of the table and finding the corresponding areas on the inside.

This paragraph presents a practical example involving bone density tests, which are measured as z-scores following a standard normal distribution. The example illustrates how to find the probability that a randomly selected adult's bone density test score is less than 1.27. The explanation includes a step-by-step guide on using both table A2 and Excel to determine the area to the left of the z-score, which corresponds to the desired probability. The paragraph also introduces the notation for probabilities and areas associated with z-scores and emphasizes the importance of understanding the problem's requirements.

The fourth paragraph demonstrates how to use Excel to calculate the probability associated with a z-score, continuing the example of the bone density test. It explains the use of the NORM.DIST function in Excel, which requires the z-score, the mean, the standard deviation, and a boolean value indicating whether the area to the left of the z-score is desired. The paragraph shows the process of entering these values into Excel and obtaining the probability that a bone density test score is less than 1.27, highlighting the increased accuracy of technology over tables.

This paragraph addresses how to find the probability that a z-score is greater than a specific value, using the example of a bone density test score being above negative 1.0. It explains that since tables only provide the area to the left of a z-score, the area to the right can be found by subtracting the area to the left from 1. The paragraph guides through the process of looking up the area to the left of negative 1.0 in table A2 and then calculating the area to the right, which represents the probability of a normal bone density test result.

The sixth paragraph shows how to use Excel to find the area to the right of a z-score, again referring to the bone density test example. It describes the process of using the NORM.DIST function in Excel to find the area to the left of negative 1.0 and then subtracting this value from 1 to obtain the area to the right. The paragraph reinforces the concept that the total area under the curve is 1 and that the area to the right is the complement of the area to the left.

The seventh paragraph discusses the calculation of the probability that a z-score falls between two values, using the example of determining the likelihood of a person having a bone density test score indicating osteopenia, which is between negative 1 and negative 2.5. It explains the process of finding the area to the left of the larger z-score and then subtracting the area to the left of the smaller z-score to find the area between them. The paragraph also shows how to perform this calculation using Excel and emphasizes the importance of understanding the concept visually rather than memorizing a formula.

The final paragraph wraps up the discussion by summarizing the process of finding areas and probabilities associated with z-scores in the context of continuous probability distributions. It reiterates the importance of understanding that areas under the curve correspond to probabilities and that the total area under the curve is always 1. The paragraph also mentions that the next video will cover how to find z-scores given known areas or probabilities, indicating a continuation of the topic in subsequent educational content.