Written Homework #2 Math& 146 N2 Please, don't write your answers on a printout of this...

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Written Homework #2 Math& 146 N2 Please, don't write your answers on a printout of this assignment, which would make them very hard to read, but use separate white (or at least very light) paper, writing with a dark pen, or, if you really insist on using pencil (which scans very badly) press it hard so that the writing is as close to deep black as possible. For your convenience, a cheat sheet and a normal table is attached. Of course, the textbook has a normal table at the end, but it doesn't extend as far as the table attached here.. Quick Cheat Sheet Reminder of Notation The "summation sign" is a very useful notation: given a set of numbers, labeled with indexes, say a, a ... "a=a+a+a+...+an i=1 For example, -1 32=12+22+32=1+4+9=14. As another example, "x Computing Variances for Sample the sample mean of a data set X1, X2, is given by The definitions of "Variance" of a data set X1, X2,..., Xare the count) 2 X X . "Sample Variance" i=1 "Population Variance" Xix2 n-1 (the mean is the sum of the data divided by Clearly 2 Sn=n n-1S2 and S=n-1 via the perfectly equivalent formula ns2.It is sometimes more convenient to compute the "population variance" "xp i=1 1*3"-" x2 ("the mean of the squares minus the square of the mean"). The sample variance can then be obtained as suggested above by multiplying this result by " n-1 Recall that the "population/sample standard deviation" is the square root of the corresponding variance. The book has a shortcut formula for the sample variance and standard deviation without explicitly going through the population variance: i=1 " n(n-1) Random Variables Sn ,"Standardizing" Consider a random variable X whose distribution has mean , and variance o. Then, the new random variable y=X- has a (different) distribution with mean 0 and variance 1. 2- 200e ox-00 1 If X has a normal distribution, that is it has a density density 2 of the form standard normal distribution, with , then I will have a 2, whose cumulative distribution function, usually denoted by D, can be read off standard tables. Example: A normal random variable has mean 2.5 and variance 9. What is the probability that an observation would turn out to be greater than 5? What is the probability that it will be less than -0.5? "Standardizing" gets us to look at the probability that a standard normal variable exceed 5-2.5 2.5 3= 30.83. That is given by 1 - D (0.83), where D is the cumulative distribution function for a standard normal variable. Looking up a table, the result is 1-0.7967-0.2033. Similarly, we will be looking for the probability of a standard normal variable to be less than- n-0.5-2.5 give you the direct answer 3=-1. A table of negative z values" will (-1)=0.1587. If you only have the usual table for positive z values, you will use the fact that P[Z 1]=1-P[Z <1]=1-0.8413, that is look up 1-0 (1) Another example: What should the standard deviation be so that X, a mean 0 normal random variable, have probability 0.999 to be no larger than 5? A standard normal variable has probability 0.999 to be less than 2.45. Hence, we look for a such that is the same as . Thus, we need 2.45=5 or. Small Tail Probabilities The tables from the book stop at a z-score of 3.59, where the probability of exceeding this number by a standard normal variable is approximately down to almost 0.0001. Most tables stop even earlier, for example at 3.09 (when the probability of exceeding is down to 0.001). The idea is that if you look even farther, the probabilities become so small as to be "practically" equal to 0. However, "practically zero" depends on the context, and it is sometimes necessary to consider even smaller probabilities (as repeatedly quoted, the Higgs Boson observation was made "at a six-sigma" level that means looking at a table extending to a value of 6 - so that the probability of a misread of the data was down to one in a billion). Software has the ability to work at these levels, and, just in case, there is a table attached to this test providing values for small tail probabilities. 1. Descriptive Statistics A data set is presented, sorted, and with some summaries: 10.867 11.738 12.151 12.472 11.028 11.783 12.156 12.482 11.062 11.784 12.156 12.484 11.242 11.843 12.161 12.490 11.336 11.844 12.175 12.538 11.369 11.858 12.192 12.595 11.468 11.890 12.194 12.607 11.476 11.934 12.215 12.634 11.499 11.980 12.230 12.692 11.577 11.991 12.242 12.835 11.602 12.036 12.263 12.935 11.634 12.057 12.311 12.988 11.671 12.067 12.321 13.012 11.704 12.087 12.337 13.055 11.713 12.115 12.372 13.086 11.715 12.131 12.416 13.183 We find the following summaries Number of data points 64 Sum of values: 774.080 Sum of their squares: 9379.510 1. Compute the sample mean, and the sample standard deviation. 2. From the table, extract the following values: Minimum, Maximum, Range 1st Quartile, Median, 3rd Quartile 2 Probability 2.1 Conditional Probabilities Suppose you have two random variables, and, with probability distributions as described in this table (for instance, the top left entry is the probability 0.3 0.2 0.1 0.4 Let. 1. Evaluate 2. Evaluate 2 Probability: Normal Distribution Physical Measures You are plugging an appliance into a wall outlet. Your appliance will burn out if the voltage is larger than 128 V, and will fail to work if it falls below 112 V. Suppose your outlet's voltage is a normal random variable with expected value (mean) = 120 V and standard deviation = 4 (if so, you should have the electrical circuitry of your house fixed!).). 1. What is the probability of your appliance burning out? 2. What is the probability of it not working because of a drop in voltage? 3. What is the probability of your appliance working properly (neither burning out nor failing for low voltage). 4. * What should the standard deviation be to ensure that the probability of your appliance working and not burning out is at least 99.99% (i.e. that the probability of voltage being too large or too small is no more than 0.0001=104, one in ten thousand). You might want to use the additional table for tail probabilities, that is attached, since most tables (including our book's) don't reach that far. Tables of the Normal Distribution Probability Content from -00 to Z Z | 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 | 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 | 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 | 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 | 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 | 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 | 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 | 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 | 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 | 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 | 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 | 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 | 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 | 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 | 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 | 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 | 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 | 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 | 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 | 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 | 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 | 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 | 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 | 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 | 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 | 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 | 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 | 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 | 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 | 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 | 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 | 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 Far Right Tail Probabilities Z P{Z to oo) | Z P{Z to oo) | Z P{Z to oo) | Z P{Z to oo} 2.0 0.02275 | 3.0 0.001350 | 4.0 0.00003167 | 5.0 2.867 E-7 2.1 0.01786 | 3.1 0.0009676 | 4.1 0.00002066 | 5.5 1.899 E-8 2.2 0.01390 | 3.2 0.0006871 | 4.2 0.00001335 | 6.0 9.866 E-10 2.3 0.01072 | 3.3 0.0004834 | 4.3 0.00000854 | 6.5 4.016 E-11 2.4 0.00820 | 3.4 0.0003369 | 4.4 0.000005413 | 7.0 1.280 E-12 2.5 0.00621 | 3.5 0.0002326 | 4.5 0.000003398 | 7.5 3.191 E-14 2.6 0.004661 | 3.6 0.0001591 | 4.6 0.000002112 | 8.0 6.221 E-16 2.7 0.003467 | 3.7 0.0001078 | 4.7 0.000001300 | 8.5 9.480 E-18 2.8 0.002555 | 3.8 0.00007235 | 4.8 7.933 E-7 | 9.0 1.129 E-19 2.9 0.001866 | 3.9 0.00004810 | 4.9 4.792 E-7 | 9.5 1.049 E-21 These tables are public domain. They are produced by APL programs written by the author, William Knight Written Homework #2 Math& 146 N2 Please, don't write your answers on a printout of this assignment, which would make them very hard to read, but use separate white (or at least very light) paper, writing with a dark pen, or, if you really insist on using pencil (which scans very badly) press it hard so that the writing is as close to deep black as possible. For your convenience, a cheat sheet and a normal table is attached. Of course, the textbook has a normal table at the end, but it doesn't extend as far as the table attached here.. Quick Cheat Sheet Reminder of Notation The "summation sign" is a very useful notation: given a set of numbers, labeled with indexes, say a, a ... "a=a+a+a+...+an i=1 For example, -1 32=12+22+32=1+4+9=14. As another example, "x Computing Variances for Sample the sample mean of a data set X1, X2, is given by The definitions of "Variance" of a data set X1, X2,..., Xare the count) 2 X X . "Sample Variance" i=1 "Population Variance" Xix2 n-1 (the mean is the sum of the data divided by Clearly 2 Sn=n n-1S2 and S=n-1 via the perfectly equivalent formula ns2.It is sometimes more convenient to compute the "population variance" "xp i=1 1*3"-" x2 ("the mean of the squares minus the square of the mean"). The sample variance can then be obtained as suggested above by multiplying this result by " n-1 Recall that the "population/sample standard deviation" is the square root of the corresponding variance. The book has a shortcut formula for the sample variance and standard deviation without explicitly going through the population variance: i=1 " n(n-1) Random Variables Sn ,"Standardizing" Consider a random variable X whose distribution has mean , and variance o. Then, the new random variable y=X- has a (different) distribution with mean 0 and variance 1. 2- 200e ox-00 1 If X has a normal distribution, that is it has a density density 2 of the form standard normal distribution, with , then I will have a 2, whose cumulative distribution function, usually denoted by D, can be read off standard tables. Example: A normal random variable has mean 2.5 and variance 9. What is the probability that an observation would turn out to be greater than 5? What is the probability that it will be less than -0.5? "Standardizing" gets us to look at the probability that a standard normal variable exceed 5-2.5 2.5 3= 30.83. That is given by 1 - D (0.83), where D is the cumulative distribution function for a standard normal variable. Looking up a table, the result is 1-0.7967-0.2033. Similarly, we will be looking for the probability of a standard normal variable to be less than- n-0.5-2.5 give you the direct answer 3=-1. A table of negative z values" will (-1)=0.1587. If you only have the usual table for positive z values, you will use the fact that P[Z 1]=1-P[Z <1]=1-0.8413, that is look up 1-0 (1) Another example: What should the standard deviation be so that X, a mean 0 normal random variable, have probability 0.999 to be no larger than 5? A standard normal variable has probability 0.999 to be less than 2.45. Hence, we look for a such that is the same as . Thus, we need 2.45=5 or. Small Tail Probabilities The tables from the book stop at a z-score of 3.59, where the probability of exceeding this number by a standard normal variable is approximately down to almost 0.0001. Most tables stop even earlier, for example at 3.09 (when the probability of exceeding is down to 0.001). The idea is that if you look even farther, the probabilities become so small as to be "practically" equal to 0. However, "practically zero" depends on the context, and it is sometimes necessary to consider even smaller probabilities (as repeatedly quoted, the Higgs Boson observation was made "at a six-sigma" level that means looking at a table extending to a value of 6 - so that the probability of a misread of the data was down to one in a billion). Software has the ability to work at these levels, and, just in case, there is a table attached to this test providing values for small tail probabilities. 1. Descriptive Statistics A data set is presented, sorted, and with some summaries: 10.867 11.738 12.151 12.472 11.028 11.783 12.156 12.482 11.062 11.784 12.156 12.484 11.242 11.843 12.161 12.490 11.336 11.844 12.175 12.538 11.369 11.858 12.192 12.595 11.468 11.890 12.194 12.607 11.476 11.934 12.215 12.634 11.499 11.980 12.230 12.692 11.577 11.991 12.242 12.835 11.602 12.036 12.263 12.935 11.634 12.057 12.311 12.988 11.671 12.067 12.321 13.012 11.704 12.087 12.337 13.055 11.713 12.115 12.372 13.086 11.715 12.131 12.416 13.183 We find the following summaries Number of data points 64 Sum of values: 774.080 Sum of their squares: 9379.510 1. Compute the sample mean, and the sample standard deviation. 2. From the table, extract the following values: Minimum, Maximum, Range 1st Quartile, Median, 3rd Quartile 2 Probability 2.1 Conditional Probabilities Suppose you have two random variables, and, with probability distributions as described in this table (for instance, the top left entry is the probability 0.3 0.2 0.1 0.4 Let. 1. Evaluate 2. Evaluate 2 Probability: Normal Distribution Physical Measures You are plugging an appliance into a wall outlet. Your appliance will burn out if the voltage is larger than 128 V, and will fail to work if it falls below 112 V. Suppose your outlet's voltage is a normal random variable with expected value (mean) = 120 V and standard deviation = 4 (if so, you should have the electrical circuitry of your house fixed!).). 1. What is the probability of your appliance burning out? 2. What is the probability of it not working because of a drop in voltage? 3. What is the probability of your appliance working properly (neither burning out nor failing for low voltage). 4. * What should the standard deviation be to ensure that the probability of your appliance working and not burning out is at least 99.99% (i.e. that the probability of voltage being too large or too small is no more than 0.0001=104, one in ten thousand). You might want to use the additional table for tail probabilities, that is attached, since most tables (including our book's) don't reach that far. Tables of the Normal Distribution Probability Content from -00 to Z Z | 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 | 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 | 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 | 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 | 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 | 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 | 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 | 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 | 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 | 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 | 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 | 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 | 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 | 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 | 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 | 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 | 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 | 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 | 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 | 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 | 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 | 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 | 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 | 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 | 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 | 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 | 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 | 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 | 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 | 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 | 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 | 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 Far Right Tail Probabilities Z P{Z to oo) | Z P{Z to oo) | Z P{Z to oo) | Z P{Z to oo} 2.0 0.02275 | 3.0 0.001350 | 4.0 0.00003167 | 5.0 2.867 E-7 2.1 0.01786 | 3.1 0.0009676 | 4.1 0.00002066 | 5.5 1.899 E-8 2.2 0.01390 | 3.2 0.0006871 | 4.2 0.00001335 | 6.0 9.866 E-10 2.3 0.01072 | 3.3 0.0004834 | 4.3 0.00000854 | 6.5 4.016 E-11 2.4 0.00820 | 3.4 0.0003369 | 4.4 0.000005413 | 7.0 1.280 E-12 2.5 0.00621 | 3.5 0.0002326 | 4.5 0.000003398 | 7.5 3.191 E-14 2.6 0.004661 | 3.6 0.0001591 | 4.6 0.000002112 | 8.0 6.221 E-16 2.7 0.003467 | 3.7 0.0001078 | 4.7 0.000001300 | 8.5 9.480 E-18 2.8 0.002555 | 3.8 0.00007235 | 4.8 7.933 E-7 | 9.0 1.129 E-19 2.9 0.001866 | 3.9 0.00004810 | 4.9 4.792 E-7 | 9.5 1.049 E-21 These tables are public domain. They are produced by APL programs written by the author, William Knight