Question

For a bell curve with mean and standard deviation , how much area is under the curve, above the horizontal?

Answer 1

Given a mean and standard deviation. We have to calculate the area under the curve. First, we will determine the z score for the particular value of x and then look up the table for the corresponding z value.

Complete step by step solution:
We are given the bell curve with $\mu = 7$ and $\sigma = 3$.
In the normal distribution, the maximum value of the bell curve is at the mean and the total area under the curve above the horizontal is always 1.

If the area of a particular location along the horizontal is required, then the z-score is calculated for that particular value using the formula $z = \dfrac{{x - \mu }}{\sigma }$

Then, the area under the curve can be determined by looking up the table value corresponding to z value.

The standard normal distribution provides the probability of z score. The shape of the normal distribution is symmetric. In this curve, about $68\%$ values lie within one standard deviation of the mean, about $95\%$ of values lie within two standard deviations of the mean and about $99.7\%$ values will lie within three standard deviations of the mean.

Hence, the area under the curve above the horizontal is one square unit.

Additional Information:
The probability density function is used to define the probability distribution of an outcome for any random variable. There are various methods to define the probability distribution function such as normal distribution, chi square, binomial distribution, Poisson distribution etc. In normal distribution, the curve in the bell shape is used to distribute the probability of the random variable.

Note: Remember that in a normal distribution curve the peak of the curve shows the mean value and the values to the left and right of the mean shows the probability of a random variable.