Question

"According to the empirical rule, we would expect approximately 68, 95, and 99.7 percent of the values to fall within one, two, and three standard deviations above and below the mean respectively for all distributions."

Answer 1

Answer:

False

Step-by-step explanation:

1) This assertive starts correctly this is called the empirical rule, and that's shows us the percent 68,95, and 99.7 to fall within one, two and three standard Deviations of the population $(\sigma)$ above and below the  mean $\mu$ (of the population, but for Normal distributions, not for binomial, Poisson, etc. distributions.

2) Check the example below

Answer 2

0Answer:

False. It applies mainly for the Normal Distribution, but not for all distributions.

Step-by-step explanation:

The empirical rule (known also as the 68-95-99.7 rule) is particularly applied to the Normal Distribution because of its unique characteristics such as its symmetry and definition by two parameters: the mean and the standard deviation.

That rule indicates that below and above the population's mean, within a distance of one standard deviation, there are 68.27% of the cases that follows this distribution; within two standard deviations, below and above, there are 95.45% of the values, and within three standard deviations 99.73% of the cases. Not all distributions have this characteristic.

Having into account such a characteristic, for populations that follow a Normal Distribution, we can standardize the values for this distribution using for this the mean and the standard deviation, ending up with a standard normal distribution in which this empirical rule is true for all data that follows a normal distribution. Because of this and that the Normal Distribution is symmetrical, we can determine probabilities for all populations that follow this distribution in a relatively easy way.

This can be achieved using the z-scores, that can be obtained using this formula:

$\\ z = \frac{x - \mu}{\sigma}$

Where x is a value for the population, $\\ \mu$ is the population's mean and $\\ \sigma$ is the standard deviation, to then consulting a standard normal table to find the corresponding probabilities.