STAT460: Elementary Probability and Statistics  
 
Final Exam
 
(Dec.13, 2000)

 

 

Student's Name (Please print):

 

SSN(Last 4 digits only):

 

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$\bullet$ There are six questions. To get full marks, a total of 30 points, you must solve five of them.

$\bullet$ Please read all six questions before you begin. You may attempt all six and the five highest scores will be taken as your mark.

$\bullet$ Solutions must exhibit a clear, complete line of reasoning in order to receive full credit.

 

Problem 1. There are 12 computers in a computer lab of the Computer Science Department, which work independently. If each computer works with probability 0.98, find the following probabilities:

(1)   At most one does not work.

(2)   At lease one works.

(3)   All work.

 

Problem 2. Suppose that a regional computer center wants to evaluate the performance of its disk memory system. One measure of performance is the average time between failures of its disk drive. To estimate this value, the center recorded the time between failures for a random sample of 60 disk-drive failures, with the sample mean is 1780 hours and sample standard deviation 210 hours.

(1)   Estimate the true mean time between failures with a 95% confidence interval.

(2)   If the disk memory system is running properly, the true mean time between failures will exceed 1050 hours. Based on the interval in (1), what can you infer about the disk memory system?

 

Problem 3. A computer company takes a simple random sample of 200 male students and another simple random sample of 300 females on WSU campus. As it turned out, 107 of the sample men used a PC computer on a regular basis, compared to 132 of the women: 53.5% versus 44%. Is the difference between the percentages real, or a chance variation? (To answer this question, you may construct a 95% confidence interval)  

Problem 4. A questionnaire was mailed to a sample of 150 households within 2 weeks after a nuclear mishap occurred in 1979 on Three Mile Island near Harrisburg, Pennsylvania. One question concerned residents' attitudes toward a full evacuation:``Should there have been a full evacuation of the immediate area?" Residents were classified according to the distances (in miles) of the community in which they reside from Three Mile Island and their opinion on a full evacuation. A summary of the responses for the 150 households randomly selected is given in the following two-way contingency table.

 

Table: Observed counts for the contingency table of Three Mile Island survey

		Distance from Three Mile Island
		1-6	7-12	13+
	yes	18	15	33
Full evacuation
	no	20	19	45

The objective of the study is to determine whether the two classifications, distance from Three Mile Island and opinion on full evacuation, are dependent.

(1) Under the hypothesis of independence between the two classifications, fill each cell of the expected cell counts for the contingency table.

 

Table: Expected counts for the contingency table

		Distance from Three Mile Island
		1-6	7-12	13+
	yes
Full evacuation
	no

(2) Calculate the $\chi^2$ statistic.

(3) Is there sufficient evidence to conclude that opinion on full evacuation is independent of distance from Three Mile Island? Use $\alpha=10\%$.

 

Problem 5. Let X and Y denote the lifetimes of two different types of components in an electronic system. The joint density of X and Y is of the form

$$f(x, y) = \left\{ \begin{array}{ll} c x e^{-(x+y)/2}, ~ & ~\... ...\ ~ & ~ \\ 0, ~ & ~\mbox{otherwise}, \end{array} \right.$$

where c is a positive constant.

(1) Find the value of c that makes f(x, y) a proper density function.

(2) Find the marginal density functions f₁(x) and f₂(y).

(3) Compute EX, EY and Cov(X, Y).

 

Problem 6. A math placement test is given to all entering freshmen at a small college. A student who receives a grade below 35 is denied admission to the regular math course and placed in a remedial class. The placement test scores and the final grades for some students who took the regular course were recorded as follows.

$$\begin{tabular}{\vert c\vert c\vert\vert c\vert c\vert} Place... ...8 \\ 90 & 79 & 65 & 57 \\ 35 & 59 & 50 & 79 \\ \end{tabular}$$

(1) A stem-and-leaf plot for the course grade is shown below. To make a comparison, construct a stem-and-leaf plot for the placement test using the same stems.

  Placement Test                           Course Grade
        |                                    1  |  1
        |                                    2  |
        |                                    3  |  6
        |                                    4  |  1  8  7
        |                                    5  |  3  6  9  4  3  7
        |                                    6  |  1  8  8
        |                                    7  |  0  9  1  1  9
        |                                    8  |
        |                                    9  |  1

(2) Find the five-number-summary for the placement test score and the course grade.

	minimum	1st quartile	median	3rd quartile	maximum
Placement
test score
Course grade

(3) Make a box plot for the placement test score and the course grade, respectively.

(4) Some summary statistics are

Sum (Placement  Test)=1110,    Sum(Course  Grade)=1173,

Sum (Placement  Test²)=67100,    Sum(Course  Grade²)=74725,

Sum (Placement  Test * Course  Grade)=67690.

Find the equation of the regression line to predict course grades from placement test scores.

(5) If 60 is the minimum passing grade, below which placement test score should students in the future be denied admission to this course?