6.2.3 Nonstandard Normal Distributions - Given areas or probabilities, find a range of x values.

This paragraph introduces the learning outcome for finding x values in a normal distribution given certain probabilities or areas. It emphasizes the importance of graphing normal distributions to visualize and work with them effectively. The speaker advises labeling the graph with any given information, such as probabilities or x values, and shading the appropriate areas. The paragraph also clarifies the difference between z-scores and areas, explaining that z-scores represent distances from the mean, while areas represent regions under the probability density function. It advises students to refer to the correct part of the table when given a z-score or area and to choose the correct side of the graph based on the problem statement. The speaker also points out common mistakes, such as confusing z-scores with areas, and the importance of ensuring that probabilities and areas are between 0 and 1.

The speaker provides a step-by-step procedure for finding x values from non-standard normal distributions, which involves sketching the normal curve, labeling given probabilities or areas, and shading the appropriate regions. They explain how to use Excel's NORM.INV function and Table A2 to find corresponding z-scores and x values. The paragraph highlights the importance of ensuring that the z-score is positive when the x-value is above the mean and negative when below, and vice versa. It also stresses the need to check that all probabilities and areas are between 0 and 1 and to use keywords in the problem statement to determine the correct side of the graph for the z-score. An example is given involving the heights of women and the U.S. Air Force pilot requirements, demonstrating how to find the 95th percentile height using the described methods.

This paragraph delves into the specifics of using Excel and Table A2 to find the x-value corresponding to a given area under the normal distribution curve. The speaker demonstrates how to use Excel's NORM.INV function to find the 95th percentile height for women, which would allow the shortest 95% to be pilots. They show the process of inputting the probability, mean, and standard deviation into the function and obtaining the x-value. The paragraph also explains how to use Table A2 to find the z-score corresponding to an area, emphasizing the need to look in the body of the table for the area and then finding the z-score on the outside edge. The speaker provides a detailed explanation of how to convert the z-score back to an x-value using a rearranged z-score formula and the importance of ensuring that the final answer makes sense in the context of the problem.

The speaker summarizes the process of converting known areas to x values in normal distributions, starting with graphing the region and finding the area to the left. They explain how to find the corresponding z-score using Table A2 by looking in the body of the table and then converting the z-score back to an x-value using the rearranged z-score formula. The paragraph reiterates the importance of checking that the final answer is contextually appropriate, ensuring that areas and probabilities are between 0 and 1, and that z-scores and x-values correspond correctly to their positions relative to the mean. The speaker concludes with an example of how changing the height requirement for U.S. Air Force pilots to 68.5 inches or less would allow 95% of women to be eligible, providing a practical application of the concepts discussed.

In the final paragraph, the speaker wraps up the lesson by emphasizing the key concept that the area bounded by a given x value in any normal distribution is the same as the area bounded by the corresponding z-score in the standard normal distribution. They explain that the process involves using the appropriate formulas to convert between a non-standard normal distribution and a standard normal distribution. The speaker also reminds learners that they can use the z-score formula to go from a standard normal distribution to a non-standard one, and vice versa. The lesson concludes with a reminder of the importance of understanding the relationship between areas, z-scores, and x-values in normal distribution analysis.