Question

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? Hint: Use Excel or Appendix C-2 to find the z-score. (Round your answer to 2 decimal places.) Standard deviation $

Answer 1

Answer:

The standard deviation is $26.67.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 157$

Assuming a normal distribution, if 70 percent of the filings cost less than $171.00, what is the standard deviation?

This means that when X = 171, Z has a pvalue of 0.7. So when Z = 0.525.

$Z = \frac{X - \mu}{\sigma}$

$0.525 = \frac{171 - 157}{\sigma}$

$0.525\sigma = 14$

$\sigma = \frac{14}{0.525}$

$\sigma = 26.67$

The standard deviation is $26.67.