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 44% of contributions. round z - value calculations and final answers to 2 decimal places. the lower and upper limits for the middle 44% of contributions are respectively $ and $. correct answer the lower and upper limits for the middle 44% of contributions are respectively $732.26 and $851.74.

Explanation:

Step1: Find the area in the tails

Since we want the middle \(44\%\), the area in the two tails is \(1 - 0.44=0.56\). The area in each tail is \(\frac{0.56}{2}=0.28\).

Step2: Find the \(z\) - values

Using the standard normal distribution table (or a calculator with a normal - distribution function), we find the \(z\) - value corresponding to an area of \(0.28\) in the left tail. The \(z\) - value \(z_1\) (lower \(z\) - value) is such that \(P(Z\lt z_1)=0.28\), and \(z_1\approx - 0.58\). The \(z\) - value \(z_2\) (upper \(z\) - value) is such that \(P(Z\lt z_2)=0.28 + 0.44=0.72\), and \(z_2\approx0.58\).

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

We know that \(\mu = 792\) and \(\sigma = 103\).
For the lower limit: \(x_1=\mu+z_1\sigma=792+( - 0.58)\times103=792 - 59.74 = 732.26\).
For the upper limit: \(x_2=\mu+z_2\sigma=792 + 0.58\times103=792+59.74 = 851.74\).

Answer:

The lower and upper limits for the middle \(44\%\) of contributions are respectively \(\$732.26\) and \(\$851.74\).