Question

Pay your bills: In a large sample of customer accounts, a utility company determined that the average number of days between when a bill wassent out and when the payment was made is 27 with a standard deviation of 5 days. Assume the data to be approximately bell-shaped.

Answer 1

17 and 37 days

Explanation:$\begin{gathered} \operatorname{mean}\text{ = }\mu\text{ =27} \\ \text{standard deviation = }\sigma=\text{5} \end{gathered}$$\begin{gathered} Since\text{ it is approx i}mately\text{ bell shaped, it will follow a normal distribution:} \\ for\text{ a normal distribution, using the empirical rule:} \\ 68\text{ \% of the data will lie within 1 standard deviation of the mean} \\ 95\text{ \% of the data will lie within 2 standard deviation of the mean} \\ 99.7\text{ \% of the data will lie within 3 standard deviation of the mean} \end{gathered}$$\begin{gathered} 95\text{ \% of the data = }P(\mu-2\sigma\text{ < x < }\mu+2\sigma) \\ \mu\pm2\sigma\text{ = }\mu-2\sigma\text{ < x < }\mu+2\sigma \\ \mu+2\sigma\text{ < x < }\mu-2\sigma\text{ } \\ \mu+2\sigma\text{ = 27 + 2(5) = 27 + 10} \\ \mu+2\sigma\text{ =}37 \end{gathered}$$\begin{gathered} \mu-2\sigma\text{ =}27\text{ - 2(5) = 27 - 10} \\ \mu-2\sigma\text{ =}17 \\ \\ \mu-2\sigma\text{ < x < }\mu+2\sigma\text{ = 17< x < }37 \end{gathered}$

Hence, approximately 95% of the customer account have payment between 17 and 37 days