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".

2) Solution to the problem

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

$X \sim N(1050,218)$

Where $\mu=1050 kWh$ and $\sigma=218kWh$

For this part we want to find a value a, such that we satisfy this condition:

$P(X>a)=0.55$ (a)

$P(X$ (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.45 of the area on the left and 0.55 of the area on the right it's z=-0.26. On this case P(Z<-0.126)=0.45 and P(z>-0.126)=0.55

If we use condition (b) from previous we have this:

$P(X$

$P(z$

But we know which value of z satisfy the previous equation so then we can do this:

$z=-0.126=\frac{a-1050}{218}$

And if we solve for a we got

$a=1050 -0.126*218=1022.532$

So the value of height that separates the bottom 45% of data from the top 55% is 1022.532.

8 0

2 years ago