Statistics for Business: Decision Making and Analysis, 3rd Edition

15.4 Manipulating Confidence Intervals

We can manipulate a confidence interval to obtain ranges for related quantities. For instance, suppose that federal regulators require a bank to maintain cash reserves equivalent to 10% of its outstanding balances. How much cash is that on average per customer? Because the 95% confidence interval for μ is [$1,520 to $2,460], the 95% confidence interval for 10% of μ (0.1 × μ) is [$152 to $246].

If [L to U] is a 100(1 − α)% confidence interval for μ, then

$$\left[c\times L\text{ to }c\times U\right]$$

is a 100(1 − α)% confidence interval for c × μ and

$$\left[c+L\text{ to }c+U\right]$$

is a 100(1 − α)% confidence interval for c + μ.

The same rule applies if the parameter is p rather than μ. More generally, you can substitute a confidence interval for a parameter ...