Question

Question:Who prepares your tax return? As part of a study on income tax compliance (Behavioral Research and Accounting, January 2015), researchers found that 37% of adult workers prepare their own tax return. Assume that this percentage applies to all U.S. adult workers. Now consider a random sample of 270 adult workers.

a. Find the probability that more than 112 of the workers prepare their own tax return.

b. Find the probability that between 100 and 150 of the workers prepare their own tax return

Short answer

1. The probability that more than 112 of the workers prepare their own tax return is zero.
2. The probability that between 100 and 150 of the workers prepare their own tax return is zero.

Step-by-step solution

Given Information

The proportion of adult workers is given by

The number of sample size is

The standard deviation of the sampling distribution is computed as

$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}} \\ =\sqrt{\frac{0.37\times 0.63}{270}} \\ =0.0017 \end{gathered}$$

So, $\hat{p}$ is normally distributed with mean of 0.37 and a standard deviation of 0.0017.

(a) Compute the probability

The probability that more than 112 of the workers is computed as

(b) Identifying the probability

The probability that between 100 and 150 of the workers is computed as

$$\begin{gathered} \left(100<\hat{p}<150\right)=p\left(\frac{100-0.37}{0.0017}<\frac{\hat{p}-p}{{\sigma }_{\hat{p}}}<\frac{150-0.37}{0.0017}\right) \\ =p\left(58605.88<z<88017.64\right) \\ =p\left(z⩽88017.64\right)-p\left(z⩽58605.88\right) \\ =1-1 \\ =0 \end{gathered}$$

Hence, the probability is 0.