Name: ______Student ID: ______Score: ______

Instructions for Mid-term as Homework:

·  Name and student ID # on the top of EACH AND EVERY PAGE!

·  Due date: Nov 11th, 2005 by 5 PM.

·  This is an individual homework assignment – no collaboration, please.

·  Show your work – to the degree possible, the problems will be graded either satisfactory (S) or unsatisfactory (U) based on if you identify the correct mathematical formula and its inputs.

·  If you choose to type your answers in this word document,

1.  Use the bold courier new font to differentiate your answers.

2.  For problem 7, use a text box over each plot with your answer.

1.  (30 points) Answer each of the following as true or false, and if false, give a short answer as to why the statement is false:

a)  The difference between permutations and combinations is if order is important.

True / False

b)  You pass as many cars as pass you on the highway. You travel the average speed.

True / False

c)  The intersection of two elementary outcomes results in another elementary outcome.

True / False

d)  The probabilities of three safety circuits operating correctly are p1, p2, and p3. The circuits combined in parallel operate 1 – (1 – p1)(1- p2)(1 – p3) correctly.

True / False

e)  f[n] = 0.5n for n > 0 is a valid PDF.

True / False

f)  P(Z < -1) = P(Z > 1) if Z is a Gaussian random variable.

True / False

g)  All uncorrelated random variables are also independent.

True / False

h)  Adding several random variables together with the same distribution tends to become chi-squared distributed.

True / False

i)  Estimator bias and standard error are separate components of MSE.

True / False

j)  Confidence intervals and hypothesis tests are statistically unrelated.

True / False

k)  Hypothesis tests are restricted to tests of the population mean and variance.

True / False

l)  Linear regression fits sample data to a straight line equation.

True / False

m)  Measurement errors correlated with the experiment order will degrade the regression coefficient estimates even if the run order is randomized.

True / False

n)  Large values of R2 and/or small P-values indicate a correct model.

True / False

o)  The breakdown of variance into portions explained by the model and those not is known as analysis of variance.

True / False

2.  (10 points) Typical parts are 10% defective while premium parts are only 5% defective. A box of the premium parts is accidentally mixed in with three boxes of the typical parts.

a)  What is the defect rate for the combined parts?

b)  If a defective part is found, what is the probability it was a premium part?

3.  (5 points) The PDF of a discrete random variable is uniformly distributed from 1 to 8. Find the mean and variance.

4.  (10 points) The error correcting code for a computer hard drive can compensate for up to 2 bit errors in every block of 4096 bits. If 1 in 10-12 bits are in error, what is the probability of a block being decoded incorrectly? DO NOT EVALUATE.

5.  (10 points) The average failure time for a bolt is 10,000 hours and found to follow an exponential distribution.

a)  How many bolts will last 5 years?

b)  How many bolts would it take to achieve an average combined failure, that is, failure of all the bolts, of at least 5 years?

PART (b) ISN’T WELL WORDED – NO NEED TO ATTEMPT.

6.  (10 points) The table below contains 20 parts measurements. The sample mean is 8.815 and the sample variance is 1.6129 for the parts.

9.6 / 7.4 / 11.6 / 11.4 / 10.2
8.0 / 11.3 / 6.8 / 9.0 / 11.7
8.1 / 7.2 / 7.9 / 9.8 / 5.5
7.9 / 8.5 / 6.4 / 8.6 / 7.3

a)  Find a 99% confidence interval on the mean assuming the data are Gaussian with unknown variance.

b)  Does the result of (a) suggest the parts population mean is 10? Why?

7.  (15 points) The joint PDF of two random variables is

f(x,y) = c x y for the range 0 < x < y < 1.

a)  Find the value of c so that f(x,y) is a valid PDF.

b)  Find f(x).

c)  Find f(y|x).

8.  (10 points) The following data table represents the results of running an experiment. Below are various plots of the regression results. Comment on each of the plots noting if the plot is benign or indicates an issue or difficulty with the experiment and its analysis.

Exp / xi / xi2 / xi yi / yi / Order / ŷi / ei / Rank / z-score
1 / 2 / 4 / -2.74 / -1.37 / 11 / -2.43 / 1.06 / 13 / 0.32
2 / 2 / 4 / -9.88 / -4.94 / 18 / -2.43 / -2.51 / 6 / -0.60
3 / 3 / 9 / -5.1 / -1.7 / 15 / -0.42 / -1.28 / 8 / -0.32
4 / 3 / 9 / -6.33 / -2.11 / 17 / -0.42 / -1.69 / 7 / -0.45
5 / 5 / 25 / 18.35 / 3.67 / 16 / 3.59 / 0.08 / 10 / -0.06
6 / 5 / 25 / 15.6 / 3.12 / 14 / 3.59 / -0.47 / 9 / -0.19
7 / 5 / 25 / 1.35 / 0.27 / 3 / 3.59 / -3.32 / 4 / -0.93
8 / 6 / 36 / 46.98 / 7.83 / 8 / 5.59 / 2.24 / 15 / 0.60
9 / 6 / 36 / 35.94 / 5.99 / 13 / 5.59 / 0.40 / 11 / 0.06
10 / 6 / 36 / 53.22 / 8.87 / 6 / 5.59 / 3.28 / 16 / 0.76
11 / 8 / 64 / 119.84 / 14.98 / 2 / 9.60 / 5.38 / 18 / 1.15
12 / 8 / 64 / 132 / 16.5 / 4 / 9.60 / 6.90 / 19 / 1.44
13 / 8 / 64 / 21.04 / 2.63 / 12 / 9.60 / -6.97 / 2 / -1.44
14 / 9 / 81 / 62.73 / 6.97 / 20 / 11.61 / -4.64 / 3 / -1.15
15 / 9 / 81 / 109.53 / 12.17 / 9 / 11.61 / 0.56 / 12 / 0.19
16 / 9 / 81 / 145.35 / 16.15 / 1 / 11.61 / 4.54 / 17 / 0.93
17 / 11 / 121 / 254.1 / 23.1 / 5 / 15.62 / 7.48 / 20 / 1.96
18 / 11 / 121 / 191.4 / 17.4 / 7 / 15.62 / 1.78 / 14 / 0.45
19 / 12 / 144 / 89.88 / 7.49 / 19 / 17.62 / -10.13 / 1 / -1.96
20 / 12 / 144 / 179.64 / 14.97 / 10 / 17.62 / -2.65 / 5 / -0.76

EMIS 7300 Fall 2005 Mid-term as Homework 1 of 9