Question

The average charitable contribution itemized per income tax return in a certain state is $792. Suppose that the distribution of contributions is normal with a standard deviation of $103. Find the limits for the middle 80% of contributions. Round z-value calculations and final answers to 2 decimal places.

Answer 1

So here we have a typical problem with a normal distribution. The curve defined by a normal distribution looks like this:

We are being asked to find the limits for the middle 80% contributors. This basically means that we'll have to find two values in the horizontal axis. The consition these two values have to met is that 0 is their mid value and the area of the curve limited by them has to be the 80% of the total area under the curve. In order to find these values we will have to use what is known as a z-values table.

Before continuing let's think about the following. We want to find an area centered around z=0 that is the 80% of the total area under the curve. This means that the areas at the left and at the right of this area must have the 10% of the total are each. The z-values table gives us the % of the area under the curve between -∞ and a given z. This means that we have to find the z-values for which this percentage is 10 and 90. So basically in the table we have to find the z's that have 0.10 and 0.90. These are those z-values in the table:

The z-values are given by the sum of the numbers in the rows and those in the columns so basically the values we are looking for are z=-1.28 and z=1.28.

In summary, the 80% of the area under the curve is equal to the are under the curve between z=-1.28 and z=1.28. However these are not the limits we are looking for. We still need to make a transformation. These z-values meet the following relation:

`z=(x-\mu)/(\sigma)`

Where μ is the mean of the normal distribution (in this case this is the the average charitable contribution that is equal to $792), σ is the standard deviation (here it's $103) and x are going to be the limits we are looking for. Then if we use the 2 z-values we found we have:

`\begin{gathered} -1.28=(x-792)/(103) \\ 1.28=(x-792)/(103) \end{gathered}`

We can multply both sides of each equation by 103 and then substract 792 from both sides:

`\begin{gathered} -(1.28\cdot103)+792=x \\ 660.16=x \\ (1.28\cdot103)+792=x \\ 923.84=x \end{gathered}`

Which means that the middle 80% contributions are between $660.16 and $923.84.