Question

Select all the properties that must be true of a normal distribution. aThe distribution is bell-shaped. bThe distribution is determined by the mean and standard deviation. cThe distribution has a height of zero when it is more than 3 standard deviations away from the center. dThe distribution is centered at the mean absolute deviation. eThe total area under the normal distribution curve is 1. fThe area under the normal distribution curve and within 2 standard deviations of the center is about 95% of the total area under the curve.

Answer 1

To answer this question, we're going to check each statement.

The first statement is 'The distribution is bell-shaped.'

This is true! The normal distribution(also named Gaussian Distribuition) is a distribution symetric in the middle, falling exponentially(with the argument squared), ending up in a "bell shape".

The second statement is "The distribution is determined by the mean and standard deviation."

This is false. With the mean and the standard deviation you can't determinate the function that results in your distribution.

To see that we can look to the definition of a gaussian distribuition

`f(x)\text{ = }\frac{1}{\sigma\sqrt[]{2\pi}}e^{(-1)/(2)(\frac{x\text{ - }\mu}{\sigma})^2}`

Where 'sigma' represents the standard deviation and 'mu' represents the MAD.

The third statement is "The distribution has a height of zero when it is more than 3 standard deviations away from the center."

It is not precisely zero, but it is almost zero. The Normal distribuition is centered at the mean and falls rapidly. Since it is not exactly zero, I'm going to say this one is false.

The fourth statement is "The distribution is centered at the mean absolute deviation."

This one is true. The distribuition is centered at the MAD.

The fifth statement is 'The total area under the normal distribution curve is 1.'

This one is true! The normal distribuition is normalized!

The last statement is 'The area under the normal distribution curve and within 2 standard deviations of the center is about 95% of the total area under the curve.'

This is also true!

The final answer would be:

True

False

False

True

True

True