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\begin{document}





\author{Brian C. Hall}

\title{Math 214--Final Exam}

\date{4:15--6:15 p.m.}

\date{4:15--6:15 p.m.}

\maketitle







\textbf{Instructions}. There are 12 problems with point-values as indicated,

with a total of 150 points possible. Answer all the questions. \textit{Show

your work on all problems.} Calculators and one 3$\times$5 inch notecard with

formulas are permitted.



This test is subject to the Notre Dame honor code.\bigskip



{\LARGE Name}:\bigskip



\begin{enumerate}

\item (12 points) Each part produced by a certain process is inspected by two

different inspectors. Experience shows that when a part is defective, 80\% of

the time the first inspector catches the defect, 80\% of the time the second

inspector catches the defect, and 70\% of the time both inspectors catch the defect.



a) Let $A$ be the event that the first inspector catches a defect in a given

defective part and let $B$ be the event that the second inspector catches the

defect. Are $A$ and $B$ independent? Why or why not?



b) What is the probability that a defective part will pass by both inspectors

without being caught?\pagebreak 



\item (12 points) A jar contains 5 red balls, 5 white balls, and 5 blue balls.



a) If you select 3 balls at random \textit{without} replacement, what is the

probability that the three balls will be of three different colors?



b) If you select 3 ball at random \textit{with} replacement, what is the

probability that the three balls will be of three different colors?\pagebreak 



\item (12 points) Suppose that $X$ is a normal random variable with mean 25.

Suppose also that $P(X>30)=0.10.$



a) Determine the standard deviation of $X.$



b) Compute $P(X>35).$\pagebreak 



\item (12 points) What is the probability of getting fewer that 80 heads out

of 200 flips of a fair coin?\pagebreak 



\item (12 points) A flat panel display contains 1,200,000 pixels. The display

is considered unusable if more than 3 pixels are defective. Suppose that the

probability of a defect on any one pixel is 1/500,000 and that the pixels are

independent. What is the probability that the display will be unusable?\pagebreak 



\item (12 points) Suppose that the average weight of tomatoes in a given field

is 3.2 ounces and the standard deviation of the weights is 0.8 ounces. The

tomatoes are packed with 40 tomatoes per crate. Let $\bar{X}$ denote the

average weight of tomatoes in a crate. 



a) Does $\bar{X}$ have an approximately normal distribution? Explain why or

why not. What are the mean and standard deviation of $\bar{X}\,$?



b) If you select a crate at random, what is the probability that the average

weight of the tomatoes in that crate will be at least 3 ounces?\pagebreak 



\item (13 points) To test the effectiveness of an SAT preparation service, 300

high school seniors are selected to participate in a study. Of these 300, 200

are selected at random to receive the preparation service, while the other 100

receive no preparation and serve as a control group. All 300 students then

take the SAT, with the following results. The average combined score among the

group receiving the preparation was 1056 with a standard deviation of 143. The

average combined score among the control group was 998 with a standard

deviation of 159.



State the null hypothesis and the alternative hypothesis for this test. Do the

results provide sufficient evidence to show that the preparation service is

effective? Test at the 1\% level.\pagebreak 



\item (13 points) A sample of 6 emeralds from a given region in Indonesia is

taken, with the following masses (in grams): 10.3, 11.2, 13.4, 15.1, 16.9,

18.0. Assuming that masses of emeralds in the population are normally

distributed, construct a 90\% confidence interval for the average mass of

emeralds in this population.\pagebreak 



\item (13 points) A ``sports betting advisor'' claims to be able to predict

the outcome of professional football games with 75\% accuracy. A government

investigator doubts that the advisor's accuracy is this high. As a test, the

investigator obtains predictions from the advisor on 110 randomly chosen games

over the course of one season. The result is that the advisor correctly

predicted the outcome of 73 of the 110 games.



Do the results provide sufficient evidence to dispute the advisor's claim?

State the null hypothesis and the alternative hypothesis for this test. Test

at the 1\% level.\pagebreak 



\item (13 points) A political candidate conducts a survey of 300 female voters

and 250 male voters in a given city. Among the female voters surveyed, 176

supported the candidate, and among the male voters surveyed, 120 supported the

candidate. Construct a 95\% confidence interval for the difference between the

percentage of female voters in the city who support the candidate and the

percentage of male voters in the city who support the candidate. Do the

results indicate that the candidate's support is higher among female voters?

Explain why or why not.\pagebreak 



\item (13 points) A theoretical model predicts that the standard deviation in

the diameters of parts produced by a certain process will be 0.0023 cm. To

test this prediction, 25 parts are produced by this process and their

diameters are measured. The standard deviation of the 25 measured lengths is

0.0016 cm. Are the results consistent with the theoretical model? State the

null and alternative hypotheses, and test at the 5\% level.\pagebreak 



\item (13 points) A study involving 18 people compares bacteria levels on the

skin after hand-washing, using a standard soap and a new anti-bacterial soap.

The participants are randomly divided into two groups, with 10 to use the new

soap and 8 to use the standard soap. The resulting measurements of bacterial

levels are as follows.

\[%

\begin{array}

[c]{ccc}%

\text{\textbf{new soap}} &  & \text{\textbf{standard soap}}\\

n_{1}=10 &  & n_{2}=8\\

\bar{x}_{1}=209 &  & \bar{x}_{2}=246.2\\

s_{1}=12.9 &  & s_{2}=9.8

\end{array}

\]

Construct a 95\% confidence interval for the average reduction in bacteria

levels for the new soap, as compared to the standard soap.



State the assumptions that one must make in order for the construction of the

interval to be valid.

\end{enumerate}

\end{document}