Question

For a normal distribution, is it likely that a data value selected at random is more than 2 standard deviations above the mean? No. This event happens only 2.5% of the time. No. This event happens only 16% of the time. No. This event happens only .015% of the time. Yes. This event happens quite frequently.

Answer 1

Answer:

$P(X>\mu + 2\sigma)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

Using the z score formula we want this probability:

$P(Z>2)$

And using the complement rule we got:

$P(Z>2) = 1-P(Z<2) = 1-0.97725= 0.023$

And that correspond to only 0.023*100= 2.3%, so is not frequently and the most appropiate answer for this case would be:

No. This event happens only 2.5% of the time

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the variable of interst of a population, and for this case we know the distribution for X is given by:

$X \sim N(\mu,\sigma)$  

Where $\mu$ the mean and $\sigma$  the deviation

We are interested on this probability

$P(X>\mu + 2\sigma)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

Using the z score formula we want this probability:

$P(Z>2)$

And using the complement rule we got:

$P(Z>2) = 1-P(Z<2) = 1-0.97725= 0.023$

And that correspond to only 0.023*100= 2.3%, so is not frequently and the most appropiate answer for this case would be:

No. This event happens only 2.5% of the time