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# Final Notes of all Lectures MAT 1000-001

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Date Created: 12/17/15

9/2/15 Lecture Notes Statistics -Study of data -Too much data! -Find Patterns -Numerical data is what we will look at Individuals -Things that have measurable attributes -Ex: -Students in this class -The 50 US states -Registered voters in the state of Michigan -Hairs on my head Variables -Measurable attributes of these individuals -The values of a variable differ among the individual -Ex: -Sex (Male/Female) [Non numerical] -Height (69 in is better than 5 ft. 9 in.) -How many hours you study a night? -How much change do you have right now? Distributions -Description of all the values taken by a variable, including repeats -Are the values similar? -Are they widespread? Are they mostly similar, with a few exceptions? Histograms -Visual representation of a distribution -Lump similar values together -Heights of bars represent the number of occurrences Notes on Histograms -No spaces between the bars -Bars have accurate heights -Some bars might have height zero -If value lies right on the boundary, then it contributes to the class on the right Interpreting Histograms -Outlier- Specific value that falls outside the overall pattern -We can describe the shape of a histogram -How many peaks? -Is it skewed right or left? -Or symmetric? 9/9/15 Lecture Notes Stemplots -Another visual representation of a distribution -Steps: 1. Round the data if necessary 2. Separate data into: a. Leaf-Last digit of each value b. Stem- Rest of the value 2 Group values with the same stem Rounding Data -If data to too precise, you won't get any grouping -Example: -8.623 8.735 9.529 9.873 10.023 -Round to 8.6 8.7 9.5 9.9 10.0 Stemplots VS Histograms -Histograms are more detailed, but harder to make -Stemplots are easier to make, but less detailed Dotplots -Yet another visual representation of a distribution -Even easier to make than a stemplot, but has less detail. -Useful to obtain a quick overview of a distribution Mean -Numerical descriptions of distributions are sometimes preferable to visual descriptions -Assign a number to the "center" of a distribution -The mean, or average of a distribution is: X=x +1 +x2+…3 n -Use the same units as the original data in your answer! -Exclude outliers before computing means Median -Mean is very sensitive to outliers -Mean value is not necessarily "typical" -Median M is the middle value -Sort the values in increasing order -M is the value in the middle of the list -If there is a tie for the middle, then take the mean of the two middle values -Median is not as sensitive to outliers 9/14/15 Lecture Notes Quartiles -One way to measure the spread 1 Sort the data 2 Find the median 3 Split the data into two halves 4 The first quartile Q1is median of the lower half 5 The third quartile Q 3s the median of the upper half Interpreting Quartiles -The median splits the data into two equal halves -The quartiles and the median split the data into four equal parts Five Number Summary -Consists of: 1 Maximum (Largest value) 2 Third quartile Q 3 3 Median 4 First quartile1Q 5 Minimum (Smallest value) Boxplots -Visual description of a five-number summary -To construct a boxplot: 1 Make a box from Q 1o Q 3 2 Draw a line through the box at M 3 Extend lines from the box to the max and the min Standard Deviation -Another numerical measure of spread -Typically accompanies the mean x -To compute the standard deviation s: x 1 Fine the mean 2 Make a table of deviations and squared deviations 3 Find the variance by "averaging" the squared deviations 4 s is the square root of the variance 9/16/15 Lecture Notes Interpreting Standard Deviation 2 -You could use |xn−´x| instead of (xn−x´), but this isn't useful -Squaring emphasizes the large deviations and suppresses the small deviations -Divide by n-1, not n, for a subtle but important reason -s=0 when all of the values are the same (no spread) -Larger s means more spread, and smaller s means less spread -The original values x and s, all have the same units Five-number Summary vs Mean and Standard Deviation -The standard deviation can't measure skew. -Mean and standard deviation are sensitive to outliers -Use the five number summary and boxplot for skewed distributions or distributions with outliers -Use mean and standard deviation for symmetric distributions with no outliers Normal Distributions -A specific type of distribution that is common in real-world data -Examples: -Biological data (heights of American females) -Standardized test scores -The "bell curve" is the same thing -Normal distributions come in many shapes Area Under Normal Distribution -The total area under a normal distribution is 100% -The area under the curve between a and b is the percentage of the values between a and b Inflection Points on the Normal Distribution -Mean occurs at the peak in the center -Also the median -An inflection point occurs where there is a change in direction of curvature -Occur at x+s∧´ x−s Quartiles of Normal Distribution -The median M is the same as the mean, at the peak of the center -Quartiles occur at: x−0.67s Q 1 Q 3 ´+0.67s 9/30/15 Lecture Notes Relationships between Variables -Each individual has two or more values -Ex: -American men: height and weight -Students at a party: beer consumption and blood alcohol concentration -The explanatory variable explains the value of the response variable -Separating cause and effect can be difficult and sometimes impossible Scatterplots -A visual representation of the relationship between two variables -The horizontal and vertical exes represent the two variables -For each individual, plot a point. Interpreting Scatterplots: Form -Scatterplots may reveal patters in data -What is the form of the scatterplots? -Linear -Non-linear -None -Are there any outliers? Interpreting Scatterplots: Direction -What is the direction of the scatterplot? -Positive association means that the two variables tend to increase together -Negative association means that one variable increases as the other variable decreases Interpreting Scatterplots: Strength -How strong is the relationship? -Nearly linear? -Vaguely linear? -No apparent relationship? Correlation -Correlation does not equal causation -r is a numerical measure of the strength and direction of a straight line relationship -r always takes a value between -1 and 1 -r = 1 means a perfect positive association -All points lie exactly on a straight line that slopes upwards -r = -1means a perfect negative association -All points lie exactly on a straight line that slopes downwards -r = 0 means there is no straight-line relationship 0<r<1 - means some positive association - −1<r<0 means some negative association Examples of Correlation Notes on Correlation -r has NO units -r does not depend on the units for the data -EX: -If you switch from inches and pounds to centimeters and kilograms, the correlation will not change at all -r measures only straight-line relationships -r is VERY sensitive to outliers 10/5/15 Lecture Notes Computing Correlation -The correlation r is not easy to compute -To compute the correlation r: x∧y´ 1 Find the means of both variables 2 Find the standard deviations sx ¿ syof both variables 3 Make a table of standard scores for both variables 4 R is the "average" of the products of the standard scores. Linear Equations -Equation of a line is: Y = mx+b -m is the slope, or rate of change -Positive slope means upward to the right -Negative slope means downward to the right -Zero slope means horizontal -b is the y-intercept -Where does the line cross the y-axis? Making Prediction with Regression Lines -You can make predictions using regression lines -If the x value for an individual is known, make an educated guess about the y value for the individual -Beware that you should NOT use a regression line to predict an x value from a known y value 10/7/15 Lecture Notes Least Squares Regression Line -The "best" linear approximation -It Minimizes the squared distance from the points on to the line Computing the Least-Squares Regression Line -The least-squares regression line is: Y = mx+b sy -m = r sx ´ -b = y-m ´,y´ ¿ -The point ( lies on the line Using Correlation and Regression -Beware of Outliers -Correlation and regression lines are very sensitive to outliers -Remove outliers before computing Correlation VS Causation -They are NOT the same thing -If two variables are correlated, it doesn't necessarily mean that one variable affects the other variable -Determining causation is both important and difficult 10/12/15 Lecture Notes Sampling -Study a large population by selecting a smaller sample -Ex: -Unemployment statistics are based on samples of the entire workforce -You can't possibly interview every U.S. resident -Chose a random sample of U.S. residents and interview them -Extrapolate from the same to the whole population Sampling Methods -Always chose a random sample -This makes the extrapolation trustworthy -Avoid bad sampling methods: -Convenience sample is based on individuals who are easy to reach -Voluntary sample is based on individuals' willingness to respond -A sample is biased if the members of the sample are not representative of the entire population -Biased sampled produce untrustworthy results Problems with Sample Surveys -Undercoverage occurs when the list of individuals is incomplete -Nonresponse occurs when some individuals refuse to participate -Wording matters -Ex: -Most people do not support "welfare", but they do support "assistance to the poor" Random Samples -Use a table of random digits to select random samples to select random samples 1 Label each individual in the population 2 Select individuals whose labels appear in the table 3 Ignore unused labels and repeated labels Experiments -A study to determine the cause-and-effect relationships -Treat some individuals and observe their responses -Decide whether the treatment causes the response Randomized Comparative Experiments -Use a randomized comparative experiment 1 Collect subjects 2 Split randomly into two groups 3 Apply different treatments to the groups 4 Compare responses Notes about Experiment's -A double-blind experiment gives particularly trustworthy results -Neither the subjects nor the experimenters know which subjects are getting which treatments -The placebo effect occurs when subjects respond psychologically to an ineffective treatment 10/14/15 Lecture Notes Parameters and Statistics -Parameter is a fixed, unknown number of interest that describes a population -Ex: -The mean income of families in Michigan -The standard deviation of heights of American males -A statistic is a number that describes a sample of a population -Ex: -The mean income of a random sample of 100 families in Michigan -The standard deviation of the heights of 1,000 randomly selected American males Inference -A conclusion about a whole population drawn from studying a sample 1 Chose a random sample 2 Compute a statistic 3 Guess (infer) the value of the parameter -Trustworthy inferences require random samples or randomized comparative experiments Sampling Distribution -How good is a guess made by inference? -What happens if you repeat a random survey many times? -Each time, the random survey will give slightly different answers -Make a histogram of these different answers -The sampling distribution is the distribution of the different values from Normal Sampling Distribution -The sampling distribution is (approximately) normal -The mean of the normal distribution is p, the parameter that we are estimating -The standard deviation of the normal distribution is: p(1−p ) s= √ n -The standard deviation s decreases as the sample size n increases -Larger surveys are more reliable Margin of Error -Recall the empirical rule ´−2s ´ +2s -For a normal distribution, 95% of the values lie between and ±2 s -Margin of error is 10/19/15 Lecture Notes Probability -The study of random outcomes -An event is random if individual outcomes are unpredictable Examples of Probability -Tossing a coin -Heads/Tails -The outcome of the next toss is unpredictable -In the long term, 50% of the tosses are heads, 50% tails -Roll a six-sided die -The outcome of the next roll is unpredictable -In the long run, 1/6 of the rolls are 4 Probability -The probability of a random outcome is the proportion (percentage) of this outcome in a very long series of repetitions -Sometimes the long series of repetitions is real, such as a roulette wheel in a casino -Sometimes the long series is hypothetical, such as the probability of a successful rocket launch Sample Spaces -A set of all possible outcomes -Ex: -Toss a coin -S = {H,T} -Roll a six-sided die -S = {1,2,3,4,5,6} Event -A subset of the sample space S -Ex: -Roll a six-sided die -S = {1,2,3,4,5,6} -Let A be the event "roll an odd number" -A = {1,3,5} -Let B be the event "roll a six" -B = {6} Probability Model -Consists of a sample space S together with numerical probabilities for each event A -Ex: -Roll a six-sided die -S = {1,2,3,4,5,6} -A = {1,3,5} (Roll an odd number) -P(A) = 1/2 Numerical Probabilities -Always between 0 and 1 -0 means impossible -1 means certain -Larger number means more likely -Use rational fractions, not decimals Complements of Events -The event that A does not occur A -P(A ) = 1 - P(A) Disjoint Events -Events A and B are disjoint if they have no outcomes in common -Can't occur at the same time -P(A or B) = P(A) + P(B) Independent Events -If the occurrence of one event does not affect the other -P(A and B) = P(A) P(B) 10/21/15 Lecture Notes Benford's Law -Applies to a list of miscellaneous numbers -Ex: -Items on a expense reports -Chose an item at random, and look at the first digit -The digits 1 through 9 are NOT equally likely -Probability: 1 2 3 4 5 6 7 8 9 .301 .176 .125 .097 .079 .067 .058 .051 .046 -This is useful for detecting fraud Equally Likely Outcomes -Many probability models consist of a finite number of equally likely outcomes -Ex: -Roll a nine-sided die. Each number one through nine is equally likely -Non-example: -Choose an item from a miscellaneous list, and look at the first digit -P(A) = Number of outcomes in A Number of outcomes in S Counting -In order to find probabilities in a situation with equally likely outcomes, you have to count things -There are several different scenarios for counting, which require different approaches "Order matters, repeats allowed" -Rule A -There are n choices -Select number of k choices -Repeats are allowed -Order matters -n = n * n * n *….. (k times) "Order matters, no repeats" Permutations -Rule B -There are n choices -There are k choices -Repeats are not allowed - P = n(n-1)(n-2)……(n-k+1) n k -Start with n, and decrease until you have a total of k numbers "Order doesn't matter, no repeats" Combinations -Rule D -There are n choices -There are k choices, in no particular order -Repeats are not allowed -nCk= n(n-1)(n-2)….(n-k+1) k(k-1)(k-2)….. -Start with n in the numerator; decrease until you have a total of k numbers -Start with k in the denominator, decrease to one 11/4/15 Lecture Notes Modular Arithmetic -Add, subtract, and multiply integers while ignoring multiplies of n -a and b are equivalent to the modulo n if their difference a-b is a multiple of n -Write a ≡ b mod n -Ex: -2 ≡ -2 6 mod 2 since both 2 and 6 are even -(-1) ≡ 1 5 mod 2 since both -1 and 5 are odd -4 ≡ 56 mod 26 since 56 - 4 = 52 = 2 x 26 Reduction Modulo n -Every integer is equivalent modulo n to 0, 1, 2, 3,… n - 2 or n - 1 -Divide a by n with remainder a = q x n + r -q is the quotient, r is the remainder -Ex: -Reduce 22 modulo 9 ≡ -22 = 2 x 9 + 4, so 22 4 mod 9 Adding, Subtracting, and Multiplying -Redice modulo n BEFORE adding, subtracting, or multiplying -a ≡ b and a' ≡ b' -Use negative numbers when convenient Solving Equations Modulo n -Apply the same algebraic techniques -Add, subtract, or multiply both sides of an equation by the same quantity -Warning: Division has some unexpected behavior Modular Arithmetic Uses -Useful for counting things that occur cyclically -Ex: -Days of the week -Months of the year -Hours of the day -We use modular arithmetic for: -Computing check digits -Encrypting and decrypting data 11/9/15 Lecture Notes Digital Information -Our world is filled with digital information -How is information stored? -How is it transmitted? -How is it encoded and decoded? -How do we know that it is correct? -How do we secure our information? Check Digits -Lots of items carry identifying numbers -Ex: -Books (ISBN) -Food (UPC) -Mail (Zip Code) -A check digit is an extra digit that helps detects errors Division-by-9 Check Digits -Used on US-Postal Service money order -First 10 digits identify the money order (unique) -The 11th digit is a check digit -To compute the check digit: a 1 a +2a + 3 + …4+ a 10 ≡ a 11mod 9 -This system can detect some errors in serial systems -It detects single-digit errors, except that: -It will not detect an error if 9 is replaced by 0 -It will not detect an error if two digits are switched -It detects the presence of an error, but it doesn't tell you how to fix the mistake(s) -If an error is detected, go back and check for mistakes Universal Product Code (UPC) -Used on all grocery products -Consists of 12 digits -Ex: 0 12345 67890 1 -The first digit represents a broad category of items -The next five identify the manufacturer -The last five identify the product -The last digit is a check digit How to compute UPC Check digits -Start with a1x a 2 a 3 a x4… + a x 11 3a 1 3a +23a + 33 + … +4a 12 ≡ 0 mod 10 UPC Check Digits -UPC check digits detect all single-digit errors -They also detect most transportation errors, where two adjacent digits are reversed except: -It doesn't detect transportations of 1,6 or 2,7 or 3,8 or 4,9 or 0,5 11/11/15 Lecture Notes International Standard Book Numbers (ISBN) -Used on all books -Two versions: 10 digits and 13 digits -We will only consider the 10 digit version -The first digit indicates the language -1 = English -Next four identify the publisher -The next four digits identify the title -The last check number is the check digit How to compute the ISBN Check Digit ≡ 10a 1 9a + 2a + 73 + 6a4+ 5a +54a + 6a + 2a7+ a 8 9 10 0 mod 11 -X counts as 10 ISBN Check Digits -Detects all single digit errors -Detects all transposition errors, where two digits are reversed -Sometimes requires the use of non-numerical X symbol Bar Codes -Written numerals can be hard for an electronic scanner -Machines have an easier time reading black and white stripes -A bar code is a series of dark bars and light spaces that encodes a number Zip Code Bar Codes -Encode Zip codes as a series of long and short bars -Each digit is encoded by 5 bars -In each block of 5 bars, two are long and three are short -Long "guard bars" start and end each bar code Zip +4 Codes -Zip +4 is a 9-digit system with more detailed address information -Encoded the same way, with longer bar codes -The tenth digit is the check digit (Only in Zip +4) a1+ a 2 a +3a + 4 + a 10 ≡ 0 mod 10 UPC Car Codes -Instead of long and short bars, information is carried in the widths of the bars and spaces -What if you scan an item backwards? -UPC bar codes use different patterns for each half 11/16/15 Lecture Notes Binary Codes -Data consisting of two different symbols -Usually 0 and 1 -Each individual symbol is a bit -Computers use binary codes internally -Zip code bar codes are binary codes: short and long bars Error Correction Schemes -Send extra information with a message -If one bit is transmitted incorrectly: -An error will be detected -An error can be fixed -This is better than a check digit, which detects errors but can't fix them -This system is important in situations where repeating the message is not practical Spacecraft Landing -NASA sends a spacecraft into Mars' orbit -Spacecraft inspects 16 possible landing sites and returns data -NASA chooses a site and sends a signal to the spacecraft on where to land -Encode the 16 sites as binary strings of length 4 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 Circle Method -Draw three partially overlapping circles -Place the four bits in the four overlapping regions -Assign 0 or 1 to the remaining regions so that each circle contains an even number of 1's -The encoded message is all seven bits Decoding with the Circle Method -Place the seven transmitted bits in the regions one through seven -Does each circle contain an even number of 1's? If yes, then the first four bits are the decoded message -If not, then change one bit so that each circle has an even number of 1's Circle Method Error Correction -Can correct at most one error -If there are two or more errors, then the Circle Method does not necessarily detect an error, and it cannot fix the error either -Only works for messages of length four 11/18/15 Lecture Notes Parity-Check Sums -Adjoin additional bits to a binary string for error correction -A number has even parity if it is even -Or, if it equals 0 modulo 2 -A number has odd parity if it's odd -Or if it equals 1 modulo 2 -Chose C t1 equal a + 1 + a 2odul3 2 C 1 0 if a 1 a + 2 is 3ven C 1 1 if a 1 a + 2 is 3dd -Chose C t2 equal a + 1 + a 3odul4 2 -Chose C t3 equal a + 2 + a 3odul4 2 Distance Between Binary Strings -The distance between two binary strings is the number of differences in their bits (Must be same length) -Ex: 101010 and 111010 are at distance one 101010 and 011011 are at distance three 101010 and 101010 are at distance zero Nearest Neighbor Decoding -Receive an encoded message, possibly with errors -Find the nearest correct message -Do not decode if there is a tie for nearest Binary Linear Codes -A set of binary strings of all possible correct messages using parity-check sums -A code word is a string in a binary code -Each code word consists of the original message with extra parity-check bits Error Detection and Correction Capacity -Find the code word with the fewest 1's (skip 0000000) -The weight t is the smallest number of 1's -The code can detect errors in at most t-1 bits -If there are more than t-1 errors, then the code may not notice a problem 11/23/15 Lecture Notes Variable-Length Code -In binary linear codes, every code word has the same length -This can be inefficient if some code words are more common than others -A variable-length code is a code in which the code word lengths are not all the same -Ex: -Morse code is a variable-length code Data Compression -Use a scheme in which the more frequently occurring data are represented by fewer bits -This can save space and make shorter transmissions -Ex: -ZIP, GIF, JPEG are data compression algorithms -Data compression is especially important for phots, videos, and music, which contain a very large amount of data Genetic Data Compression -Genes are sequences of the four nucleotides A, T, G, and C -Ex: AAACAGTAAC -A is most common, C is next most common, and T and G are slightly rarer -Encode gene sequences are: A = 0 C = 10 T = 110 G = 111 Gene Uncompressing -To decode a binary string to a sequence: -Work from left to right -Break the binary string into code words 0, 10, 110, and 111 -Convert each code word to A, C, T, and G Huffman Coding -Huffman codes are efficient for encoding data composed of symbols with variable frequencies 1) List the symbols with the leas probable first 2) Merge the two least probable symbols, with the least probable on the left 3) Reorder the list, and repeat 4) Construct a tree that represents the merging process 5) Label the branches with 0s and 1s 6) Determine code words for each symbol from the tree 11/30/15 Lecture Notes Cryptography -Encode the message so that eavesdroppers cannot decode -Only the intended receiver knows how to decode -Applications include: -Politics/military -Banking and credit card information -Personal medical information -Business Caesar Cipher -Start with plain text message -Encrypt message by replacing each letter with the letter that is three steps late in the alphabet -For X, Y, and Z, return to the start of the alphabet -Decrypt a message by replacing each letter with the letter that is 3 steps earlier in the alphabet Caesar Cipher and Modular Arithmetic -Uses arithmetic modulo 26 -Assign each letter to a number: -A = 0 -B = 1 -C = 2 … -Y = 24 -Z = 25 -Encrypt a message by adding 3 modulo 26, then convert it back to letters -Decrypt a message by subtracting 3 modulo 26, then converting back to letters Decimation Cipher Encryption -Chose a key k -An odd number from 3 to 25, except for 13 -k cannot have any common factors with 26 (2 x 13 = 26) -Convert letters to numbers (like above) Decimation Cipher Decryption -Use the same process, with a different key k -If k is the encryption key, then chose a decryption key d 12/2/15 Lecture Notes Linear Cipher -Chose a key k -Any odd number between 3 to 35, except for 13 -Chose a shift s -Any number between 0 to 25 -Convert letters to numbers -Replace i with (k * i + s) mod 26 -Multiply by k -Then add s -Reduce by modulo 26 Linear Cipher Decryption -Similar process, but with a different key -If k is the encryption key, then chose a decryption key d with a property that k * d ≡ 1 modulo 26 -Use Cipher Key table to find d -Decrypt i as d(i - s) mod 26 -First subtract the shift s -Multiply by the decryption key d -Reduce modulo 26 Vingenere Cipher -Select a keyword -The letters of the original message are shifted by different amounts -Repeat the keyword as often as necessary to encrypt longer messages Vingenere Cipher Decryption -Same process, just subtract shifts

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