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# Week 4 AMS 5

UCSC

GPA 3.24

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This 4 page Class Notes was uploaded by Sandy Nguyen on Thursday October 22, 2015. The Class Notes belongs to AMS 5 at University of California - Santa Cruz taught by Prof. Bruno Mendes in Fall 2015. Since its upload, it has received 44 views. For similar materials see Statistics in Applied Mathematics at University of California - Santa Cruz.

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Date Created: 10/22/15

Chapter 5 Normal Approximation Wednesday October 21 2015 350 PM Normal Curve Can often be used as an 39ideal39 histogram To which histograms for data can be compared Discovered in 1720 by Abraham de Moivre Also referred to as Gaussian curve or bell curve It is symmetric about zero Total area under the graph is equal to 100 gt Area under a normal curve between 1 amp 1 is 68 gt Area under normal curve between 2 amp 2 is 95 Areas under normal curve can be found by using the table at the back of the textbook Blunt11 mu quot21039qu uvb Standard Units Many histograms are similar in shape to the normal curve as long as they are drawn to the same scale A histogram and a normal curve match up via standard units A value is converted to standard units by finding how many standard deviations SDs it is above or below the average gt Values above average are positive gt Values below average are negative H 68 amp 95 Rule see figures above For many lists 68 of the entries are between average SD and average SD gt Convert interval to standard units 1 to 1 gt Area under normal curve between 1 amp 1 is 68 gt If the histogram follows the normal the area under the histogram is approx 68 For many lists 95 of the entries are between average 2 SDs and average 2 SDs gt Convert the interval to standard units 2 to 2 gt Area under the normal curbe between 2 and 2 is 95 gt If the histogram follows the normal curve the area under the histogram is about 95 Normal Approximation for Data Normal curve can be used to estimate the percentage of entries in an interval Normal approximation 1 Convert the interval to standard units 2 Find the corresponding area under the normal curve Percentiles another way summarize data The kth percentile is a number such that k of the entries in a list are smaller than the number gt 100k are larger Examples gt lst percentile number such that 1 of the entries are smaller than the number and 99 are larger gt 25th percentile number such that 25 of the entries are smaller than the number and 75 are larger Percentile rank the percentage of entries smaller than that value gt The percentile rank of the highest homework score is 100 gt The percentile rank of the median homework score is 50 A percentile is a number A percentile rank is a Estimating percentiles If a histogram follows the normal curve the normal curve can be used to estimate percentiles 1 Sketch a normal curve and find the correct quot2 valuequot by using the normal table at back of textbook 2 quotzquot is given in standard units convert it back to the units in the problem Qua rtiles lst quartile number such that 14 of the data are smaller and 34 are larger 25th percentile 2nd quartile number such that 24 of the data are smaller and 24 are larger 50th percentile median 3rd quartile number such that 34 of the data are smaller and 14 are larger 75th percentile Interquartile Range IQR another measure of the spread of the data gt IQR 3rd quartile 1st quartile Changes of Scale Adding a constant to all entries of a list gt Average increases by this constant gt Standard deviation and standard units don t change Multiplying all entries in a list by a positive number gt Average is also multiplied by this number gt Standard deviation is multiplied by this number gt Standard units don t change Multiply all entries on a list by a negative constant gt In standard units the sins are reversed Changes of scale don39t change standard units only units change Chapter 6 Measurement Error Monday October 12 2015 1 122 AM CHANCE ERRORS In reality results are thrown off by chance error gt The error changes from measurement to measurement No matter how careful a measurement could have come out slightly differently If the measurement is repeated it will come out slightly differently gt Replicating the measurement shows the difference The standard deviation of a series of repeated measurements estimates the likely size of the chance error in a single measurement gt Individual measurement exact value chance error The chance error throws each individual measurement off the exact value by an amount that changes from measurement to measurement Variability in repeated measurements reflects the variability in chance errors both gauged by the standard deviation of the data Mathematically standard deviation of chance errors must equal the standard deviation of the measurements adding the exact value is just a change of scale 0 U T L E R S In careful measurement work a small percentage of outliers is expected Upon seeing an outlier investigators chose to either ignore it or concede that their measurements don39t follow the normal curve gt lst choice is more usual triumph of theory over experience B A 5 Systematic error Affects all measurements in the same way gt Pushing measurements in the same direction Chance errors change from measurement to measurement gt Sometimes up amp sometimes down The basic equation must be modified when each measurement is altered by bias and chance error gt Individual measurement exact value bias chance error No bias in a measurement procedure the longrun average of repeated measurements should give the exact value of the thing being measured chance errors should be cancelled out Bias present the longrun average will itself be either too high or too low Usually bias can39t be detected byjust looking at the measurements themselves Measurements must be compared to an external standard or to theoretical predictions 5 U M M A R Y Chance error no matter how carefully made a measurement could have turned out slightly differently gt Estimate the likely size of chance error by replicating the measurement The likely size of chance error in a single measurement can be estimated by the SD of a sequence of repeated measurements made under the same conditions Bias systematic error causes measurements to be systematically too high or too low gt Individual measurement exact value bias chance error gt Chance error changes from measurement to measurement gt Bias stays the same amp cannot be estimated just by repeating the measurements Even in careful measurement work small percentage of outliers can be expected The average and standard deviation can be heavily influenced by outliers gt Histogram won39t follow the normal curve well Chapter 8 Correlation Thursday October 15 2015 First correlation diagram showed the relation between head circumference and height Sir Francis Galton 1822 1911 The Scatter Diagram 603 PM Scatter Diagram Illustrates the relationship between two variables Association Dependent and Independent Variables Summarizing the Data If there39s strong association between two variables knowing one helps predicting the other If there39s weak association between two variables knowing one variable doesn39t help much in guessing the other Association is NOT causation Independent variable influences the dependent variable Pomt Of allgages Depends on data Independent variable is put on the horizontal x axis Dependent variable is put on the vertical y axis Point of averages shows the average of the x values and the average of the y values o It locates the center of the point cloud Measuring the spread of the cloud from side to side use the SD of the x values the horizontal SD and the SD of the y values the vertical SD Lightly clustered scatter points strong linear association Loose clustered scatter points weak linear association The average amp SD of the x values The average amp SD of the y values Correlation coefficient r Correlation Coefficient Correlation coefficient r measure of linear association or clustering around a line 0 Quantity measuring the strength of the linear association between the two variables Calculating Correlation Coefficient 1 2 gt Convert each variable into standard units Taking the average of the products r average of x in standard units x y in standard units Interpreting Correlation Coefficient Correlations area always between 1 amp 1 and can take any value within that range Positive correlation cloud slopes up as one variable increases so does the other direct relationship Negative correlation cloud slopes down as one variable increases the other decreases inverse relationship If quotrquot is closer to 1 or 1 the linear association between the variables is stronger and the points are more tightly clustered together on the scatter diagram If quotrquot is closer to O the linear association between the variables is weaker and the points are more loosely clustered on the scatter diagram

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