Question

itemized charitable contributions 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 78% of contributions. round z - value calculations and final answers to 2 decimal places. the lower and upper limits for the middle 78% of contributions are respectively $\square$ and $\square$

Explanation:

Step1: Find the area in the tails

Since we want the middle \(78\%\) (\(0.78\)) of the distribution, the area in the two tails is \(1 - 0.78=0.22\). The area in each tail is \(\frac{0.22}{2}=0.11\).

Step2: Find the \(z\) - scores

We look for the \(z\) - score such that the area to the left is \(0.11\) and \(1 - 0.11 = 0.89\). Using a standard normal table or a calculator with a normal - distribution function (e.g., \(z=\text{invNorm}(0.11)\) and \(z=\text{invNorm}(0.89)\)), we find \(z_1\approx - 1.23\) and \(z_2\approx1.23\)

Step3: Use the \(z\) - score formula \(x=\mu+z\sigma\)

Given \(\mu = 792\) and \(\sigma = 103\)
For the lower limit (\(z=-1.23\)):
\(x_1=\mu+z_1\sigma=792+(- 1.23)\times103=792-126.69 = 665.31\)
For the upper limit (\(z = 1.23\)):
\(x_2=\mu+z_2\sigma=792 + 1.23\times103=792+126.69=918.69\)

Answer:

The lower and upper limits for the middle \(78\%\) of contributions are respectively \(\$665.31\) and \(\$918.69\)