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Taya2010 [7]
3 years ago
13

In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean

of 1050 kWh and a standard deviation of 218 kWh. Find P45, which is the consumption level separating the bottom 45% from the top 55%
Mathematics
1 answer:
Otrada [13]3 years ago
8 0

Answer:

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.  

Step-by-step explanation:

1) 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".  

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.  

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