Assume that the variable under consideration has a density curve. It is given that 34.4% of all possible observations of the variable are less than 20.
a. Determine the area under the density curve that lies to the left of 20.
b. Determine the area under the density curve that lies to the right of 20.
Answer (1)
The area to the left of 20 represents 34.4% of the observations.
Convert the percentage to a decimal: 34.4% = 0.344 .
The area to the right of 20 is calculated by subtracting the area to the left from 1: 1 − 0.344 = 0.656 .
The area to the left of 20 is 0.344 and the area to the right of 20 is 0.656 .
Explanation
Area to the left of 20 The problem states that 34.4% of all possible observations are less than 20. In a density curve, the area under the curve to the left of a certain value represents the proportion of observations less than that value. Therefore, the area to the left of 20 is 34.4% expressed as a decimal.
Calculate area to the left To find the area to the left of 20, we convert the percentage to a decimal: 34.4% = 0.344
Area to the right of 20 The total area under a density curve is always equal to 1. To find the area to the right of 20, we subtract the area to the left of 20 from the total area (1).
Calculate area to the right The area to the right of 20 is calculated as: 1 − 0.344 = 0.656
Final Answer Therefore, the area under the density curve to the left of 20 is 0.344, and the area to the right of 20 is 0.656.
Examples
Density curves are used in many real-world applications, such as analyzing exam scores or predicting customer behavior. For example, if we know that 34.4% of students scored less than 70 on an exam, we can say that the area under the density curve to the left of 70 is 0.344. This allows us to make predictions about the distribution of scores and compare the performance of different groups of students. Similarly, in marketing, density curves can help understand the distribution of customer spending, allowing businesses to target specific customer segments more effectively. Understanding areas under density curves is a basic tool for statistical analysis.
