13. suppose that form a random sample from a distribution for which the p. d.f. is f (x|θ ), where the value of the parameter θ belongs to a given parameter space ? . suppose that t = r() and t ? = r? () are two statistics such that t ? is a one-to-one function of t ; that is, the value of t ? can be determined from the value of t without knowing the values of , and the value of t can be determined from the value of t ? without knowing the values of . show that t ? is a sufficient statistic for θ if and only if t is a sufficient statistic for θ.

At river bank, checking account customers earn 0.75% interest a month on their account balances, pay no service fees for writing checks, and pay a monthly $2.00 financial charge. calculate the income earned in one month on the checking account for a customer with an account balance of $1,000. a. $4.25 b. $5.50 c. $6.75 d. $7.25 checking account earnings at baker’s bank are expressed by the equation: i = -0.09x + 10.2. while the earnings at elite bank are modeled by: i = -0.02x + 7.5. in both cases, x is the number of checks written. at what number of checks will elite bank start generating more checking account income than baker's bank? a. 12 b. 25 c. 39 d. 42 emilio’s checking account had a balance of 728.32 at the beginning of the week. he wrote checks for the following amounts throughout the week: $99.48, $33.50, $18.23, and $72.05. he also deposited his paycheck in the amount of $1,109.90. what is his account balance at the end of the week? a. $1,192.35 b. $1,222.86 c. $1,540.52 d. $1,614.96