The average cost of an IRS Form 1040 tax filing at Thetis Tax Service is \$157.00. Assuming a normal distribution, if 70 percent of the filings cost less than \$171.00, what is the standard deviation?

Solution

Let $X$ denote the cost of an IRS form 1040 tax filing at Thetis Tax Service.

$X\sim N(157,\sigma^2)$.
Given that $\mu = 157$ and $P(X < 171)= 0.7$.

Let $\sigma$ be the standard deviation.

$$ \begin{aligned} & P(X < 171) =0.7\\ &\Rightarrow P\big(\frac{X-\mu}{\sigma} < \frac{171-\mu}{\sigma}\big)=0.7\\ &\Rightarrow P(Z < \frac{171-157}{\sigma}\big)=0.7\\ &\Rightarrow \frac{171-157}{\sigma}= 0.524\\ &\Rightarrow \sigma = \frac{171 - 157}{0.524}\\ &\Rightarrow \sigma = 26.718 \end{aligned} $$
The standard deviation is $\sigma=26.718$.