Question

On a 95-degree day with 80% humidity, the amount of time it takes an air-conditioning unit to cool a 2000-square-foot home 1 degree follows a normal distribution with mean time of 24 minutes and standard deviation of 1 minute. What is the probability it takes an air-conditioning unit longer than 25 minutes to cool the house 1 degree? (Use the Empirical Rule.)

Answer 1

Answer:

In statistics, the empirical rule, also called 68-95-99.7 rule, establishes that in a normal distribution, almost (because the total is assume as 99.7%) all data can be studied into three intervals which are obtained by the standard deviation operation with the mean. The images attached show this three areas percentage where data falls, using the standard deviation.

Those numbers mean that a 68% of all data falls under the center zone, 95% falls under the next bigger zone, and a 99.7% of all data falls under the biggest area of the graph.

As you can see in the graph, the major number of data is placed in the center, inside the 68% zone, meaning that there's present the major probability to fall a single data. On the other hand, to construct each height, we must sum and subtract the deviation from the mean, if we sum this deviation, we will obtained the right heights, if we subtract the deviation from the mean, we will obtained the left heights.

So, the starting point of the normal distribution graph is the center, where the mean is placed (mean = 24 min), then we sum the deviation (deviation = 1 min) to it (24+1 = 25), therefore, the first height on the right side is 25. Then, we repeat this process, to build the rest.

Right side:

• First height: 25 min.
• Second height: 26 min.
• Third height: 27 min.

We are gonna study the right side only, because the question is about the probability to happen a longer time than 25 minutes.

Returning to the images, we are going to consider the area major than 25 minutes, that is, the area after the first heigh (right side), which is coloured by purple and green. To calculate the percentage of that area we only need to subtract 99.7% - 68% = 31.7%, then we divide it by two, because we only need the right side, having as a result 15.85%.

Therefore, the studied zone which is more than 25 minutes have a representation of 15.85%, this means that there's an almost 16% of chances that an air-conditioning unit takes longer than 25 minutes.