The state tourism commission wants to estimate the average amount out-of-state tourists spend when traveling. The commission wants to be 90% sure than their estimate is within \$60 of the true mean expenditures. A pilot sample reveals a standard deviation of \$440. How large a sample should they take? (please express your answer using 1 decimal place)


Given that the sample standard deviation $s =440$, margin of error $E =60$. The confidence coefficient is $1-\alpha=0.9$. Thus $\alpha = 0.1$.

Z-critical 0.1
Z-critical 0.1

The critical value of $Z$ is $z=Z_{\alpha/2} = 1.645$.

The minimum sample size required to estimate the mean is

$$ \begin{aligned} n &= \bigg(\frac{z* s}{E}\bigg)^2\\ & = \bigg(\frac{1.645*440}{60}\bigg)^2\\ & =145.524\\ &\approx 146. \end{aligned} $$
Thus, the sample of size $n=146$ will ensure that the $90$% confidence interval for the mean will have a margin of error $60$.

Further Reading