Answer
The properties of the normal density curveare are as follows:
1. The normal density curve is symmetric, and the mean, median, and mode occur at the same point. The mean is the highest point of the curve.
2. The inflection points of the normal distribution are \[\mu -\sigma \]and \[\mu +\sigma \], where \[\mu \]is the mean and \[\sigma \]is the standard deviation of the distribution.
3. The area under the normal curve is equal to 1. The area left to the mean value and the right to the mean value is equal and the area left to mean and right to mean is equal to 0.05.
4. The curve never reaches to the horizontal axis for any larger or smaller value of the variable.
5. The empirical rule indicates that 68% of the area under the curve is between\[\mu -\sigma \text{ and }\mu +\sigma \]; 95% and 99.7% of the area under the curve is between\[\mu -2\sigma \text{ and }\mu +2\sigma \] and\[\mu -3\sigma \text{ and }\mu +3\sigma \].
Work Step by Step
The normal distribution is a symmetric distribution, and the mean, median, and mode of the distribution are same. The inflection points of the distribution are those points where the curvature of the graph is changed. The mean divides the area under the curve in two equal parts, and the value of each part is 0.05. According to the empirical rule, 68% of the area under the curve is between\[\mu -\sigma \text{ and }\mu +\sigma \];95% and 99.7% of the area under the curve is between \[\mu -2\sigma \text{ and }\mu +2\sigma \] and \[\mu -3\sigma \text{ and }\mu +3\sigma \].
