(Solved): 2) Sampling Distribution for the Difference of Two Means a) A car manufacturer outsources produ ...

2) Sampling Distribution for the Difference of Two Means a) A car manufacturer outsources production of a certain automotive component. The company has found a cheaper supplier (supplier 2), but wants to make sure that the component is of the same quality. The component from the original supplier (supplier 1) has a lifetime of ${?}_1=50,000$ miles. The standard deviations in component lifetimes are known to be ${?}_1=$ 2,000 miles and ${?}_2=2,500$ miles, respectively. 50 components from each supplier are tested and the average lifetime for the new supplier's component is 1,050 miles shorter compared to the original supplier's component (that is, ${\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2=1,050$ ). Can the manufacturer confidently switch to the new supplier? (That is, is it reasonable to find a difference of lifetimes as great (or greater) than 1,050 miles if the lifetimes of the two components are truly equal?) i) How likely is it to find samples as extreme as (or more extreme than) these? (That is, what is $P\left({\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2>1,050\right)$, under the assumption that ${?}_2=50,000$ ?) ii) Does the assumed value of ${?}_2=50,000$ miles seem correct? (That is, is it likely that the population means are actually equal if the manufacturer finds ${\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2$ as large as 1,050 ?) Circle YES or NO and briefly explain your answer. b) A paint company has developed a new paint formula which should have a faster drying time compared to the old formula. Paint drying time is known to be normally distributed, and from previous tests it is known that the original paint formula has a mean drying time of ${?}_1=45\min \left({?}_1=6\min \right)$. The company's R\&D scientists believe the new formula will have a mean drying time of ${?}_2=35\min \left({?}_2=4\min \right)$ (that is, they expect the new formula to dry an average of $10\min$ faster than the old formula, or that ${?}_1?{?}_2=10$ ). A scientist takes 16 samples of each paint formula and determines the mean drying time for each batch to be ${\stackrel{?}{X}}_1=44\min$ and ${\stackrel{?}{X}}_2=36\min$, respectively. Does the assumed value of ${?}_2=$ 35 min seem correct? (That is, does ${?}_1?{?}_2=10$ min seem correct if he finds ${\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2=8$ ?) i) How likely is it to find samples as extreme as (or more extreme than) these? (That is, what is $P\left({\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2<8\right)$, under the assumption that ${?}_2=35$ ?) ii) Does the assumed value of ${?}_2=35\min$ seem correct? (That is, is ${?}_1?{?}_2=10$ likely to be correct if he finds ${\stackrel{?}{X}}_1?{\stackrel{?}{X}}_2=8$ ?) Circle YES or NO and briefly explain your answer.