your response &&&&&&&&&&&&&&&&&&
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mean of 30 intervals: 1.101 standard deviation:0.06493 how obtained: For my calculations listed in the last question, the mean was the sum of the 30 intervals divided by 30, to get the standard deviation I took the sum of the squared deviations of my 6 samples, averaging this sum by dividing by 5, since i had less than 30 deviations. To get the data from the program, I copied the data from the TIMER program into excel 2010, then copied the column of interval times (30 intervals) into the Data Program (downloaded program, running on laptop), and clicked on the mean and standard deviation button. Copy of data set (copied from Data Program window): 1.03125 1.152344 1.09375 1.046875 1.105469 1.136719 1.09375 1.21875 1.058594 1.140625 1.042969 1.0625 1.027344 1.140625 1.1875 1.105469 1.140625 1.027344 1.03125 1.183594 1.1875 1.011719 1.234375 1.058594 1.125 1.140625 1.136719 1.015625 1.105469 0.984375 There are 30 numbers in your distribution. Their sum is 33.03 Their mean is 33.03/ 30 = 1.101. The (mean) average deviation from the mean is 0.05512. The sum of the squared deviations is .1265. The standard deviation is therefore sqrt( .1265/ 30) = 0.06493. #$&* Investigate 'first differences' of 30-interval data Now restore your original 30 time intervals to the box. You will have to do this manually, clearing the contents of the box and then copying and pasting the data from the text editor or word processor where you stored it before. Make sure your data also stays in that location, because you'll need it at least once again. Click on the First Difference button. You will see a report of the differences between your successive time intervals. •Give the first three differences below, in the first line in comma-delimited format. •Starting at the second line answer the two questions: Are all the differences between your time intervals all different, or do some occur more than once? Where have you see this information before and what does it mean?your response &&&&&&&&&&&&&&&&&&
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first three differences of clock times:.1211,-.05859,-.04688 are differences all the same:no where seen before:these numbers are similar to the deviations seen in the Button click TIMER program exercise. #$&* Sum your 30 time intervals and speculate on meaning Restore your original 30 intervals to the box. Click on the Running Sum button. •Scroll down and take a quick look at the entire report. •Give your first three running sums below, in the first line in comma-delimited format. •Starting at the second line, explain how you think these numbers were calculated from the time intervals, and what these numbers might mean.your response &&&&&&&&&&&&&&&&&&
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running sum of intervals, first three sums:1.031,2.184,3.277 explanation of how running sums were obtained:The running sums were obtained by adding the time interval numbers to each other in a running sum. #$&* Analyze the first difference of the running sums, and the first difference of this result Delete everything but the single-column report of the running sums, so the data box contains just the running sums with one sum on each line, and click on the 'first difference' button. •Report your first three new numbers below, in the first line in comma-delimited format. •Describe what you see and what might be the meaning of the new numbers. •Suggestion: look at your original 30 time intervals. •How do you think the new numbers were calculated, where have you seen them before, and why do they come out the way they do?your response &&&&&&&&&&&&&&&&&&
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first difference of running sums, first three numbers:1.153,1.093,1.047 description and meaning: Since the running sum is calculated by adding each iteration of the time interval to each other, the first difference of the running sums is equal to each respective time interval measurement. explanation of how numbers were calculated:The first difference of the running sums is the difference between each iteration of the running sum. #$&* Again isolate only the single-column report and again click on First Differences. •Report your first three new numbers below, in the first line in comma-delimited format. •How do you think the new numbers were calculated, where have you seen them before, and why do they come out the way they do?your response &&&&&&&&&&&&&&&&&&
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first three numbers from first difference:0.06000,-.046,0.05800 how calculated:The first difference of the first difference of the running sums is calculated by taking the difference between subsequent first differences, so here we end up with values equal to the difference between clock time intervals. #$&* Find difference quotients for a new set of data and speculate on the meaning of the difference quotient Clear the box then copy the following 4 lines into the textbox: 0, 0 10,10 20,25 30,45 •Click on the Difference Quotient button. •Report below the three new numbers you see, reporting your numbers in the first line in comma-delimited format. •In the second line speculate on how the program might have calculated these numbers.your response &&&&&&&&&&&&&&&&&&
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difference quotient from sample data:1,1.5,2 how you think the results were calculated: slope of line running through consecutive points? #$&* The information in the table 0, 0 10,10 20,25 30,45 represents the position of an object rolling down an incline vs. clock time, with position in meters and clock time in seconds. Recall that according to our 'y vs. x' convention, in a position vs. clock time table the clock time is in the first column. •How far did the object travel in the first time interval? How much time elapsed while it traveled through this distance? What therefore was its average speed during this time interval? Report your numerical answers to these three questions in the first line below, in comma-delimited format. •Answer the same questions for the second time interval, and report in the second line, using the same format as in the first. •Answer the same questions for the third time interval, and report in the third line, using the same format as in the first. •Starting in the fourth line, explain how you obtained your results. •Then explain once more what the 'difference quotient' operation does to two columns of numbers.your response &&&&&&&&&&&&&&&&&&
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distance during first interval: 10m elapsed time while traveling this distance:10s average speed during interval:1meter per second 10m,10s,1meter per second distance during second interval:15m elapsed time while traveling this distance:10s average speed during interval:1.5meters per second 15m,10s,1.5meters per second distance during third interval:20m elapsed time while traveling this distance:10s average speed during interval:2meters per second 20m,10s,2meters per second explanation: I obtained the results by taking the distance traveled (which was the difference bewteen each consecutive y value, 10,15,20 in this example), and divided that by the time interval (the difference of each consecutive x value, which was 10 each time in this case) to get the speed. explanation of 'difference quotient' operation: the difference quotient is the difference in y values divided by the difference in x values, similar to the slope of a line between two given points #$&* Select and analyze 5 random intervals from 30-interval data, using the data program to find mean and standard deviation Using a coin according to the following instructions, you will now select 5 intervals randomly from your 3-interval data. You will do this by generating 5 numbers corresponding to the numbers of your data point. The process should take only a couple of minutes: Using the coin you will generate a series of numbers between 0 and 31. Note that there are 32 numbers between 0 and 31. This process can generate 32 possible numbers. If you generate a number you have generated before you will discard it and generate an alternative. If you generate a number that does not correspond to one of your intervals (probably 1-20 or 1-19) you will discard that number. You will continue until you have generated 5 numbers that haven't been discarded. To generate each number will require 5 flips of your coin. You will write down 5 numbers. Your first flip is worth 1. Flip the coin. If you get Heads write down the number 1. If you get tails write down 0. Whichever number you write down will be at the top of a column. Your second flip is worth 2. Flip the coin a second time. If you get Heads write down the number 2. If you get tails write down 0. This number does in the column below the previous. The third, fourth, and fifth flips are respectively worth 4, 8 and 16 on Heads, 0 if you get Tails. You should now have five numbers in your column. Add them up. The result will be not less than 0 + 0 + 0 + 0 + 0 and not more than 1 + 2 + 4 + 8 + 16 = 31. Go ahead and generate your first number according to these instructions. If the number is between 1 and the number of intervals you observed (e.g., between 1 and 30, or between 1 and 29), circle the number. Now generate another number, using the same procedure with 5 flips of the coin. If this number is between 1 and your number of intervals (e.g., between 1 and 30), and if it does not duplicate the first number you generated, circle it. Continue this process, generating totals between 0 and 31 and circling those that lie in the correct range and do not duplicate any your previous numbers. Stop when you have generated 5 distinct numbers within the appropriate range. Now select the time intervals corresponding to the numbers you have generated (e.g., if you had a 30-interval set and your numbers were 23, 8, 11, 19, 5 and 22 you would select the 23d, 8th, 11th, 19th, 5th and 22d time intervals). •Clear, then put these 5 time intervals into the textbox. Note that you will put time intervals into the textbox, not the numbers you have generated between 0 and 31. •Click on the Mean and Standard Deviation button. In the first line below, report the five random numbers you generated, in comma delimited format. In the second line below, report the five time intervals you put into the box, in comma delimited format. In the third line, report the mean and the standard deviation in comma-delimited format. Starting in the fourth line give a brief explanation of what your numbers mean and they were obtained. Optional comments may be added.your response &&&&&&&&&&&&&&&&&&
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your five random numbers: 14,23,28,18,11 your five intervals: 1.140625,1.234375,1.015625,1.027344,1.042969 mean and standard deviation:1.092,0.09361 explanation: The mean deviation of the random sample is the sum of the differences divided by the numer of samples, in this case the sum divided by 5. The standard deviation is the square root of the quantity of the sum of the squared deviations divided by one less than the number of samples (since this is less than 30 deviations). #$&* In three lines report the following numbers: By how much does the mean of your 5-interval sample differ from the mean of the entire data set of 30 intervals? What is the standard deviation of the 30-interval set? What is the first number you reported as a percent of the second. That is, what is the difference between your sample and the entire data set, as a percent of the standard deviation of the data set? Starting in the fourth line give a brief explanation of what your numbers mean and how you obtained themyour response &&&&&&&&&&&&&&&&&&
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difference of means:0.008 standard deviation of 30-interval set:0.06493 first number as percent of second:12.3% explanation and meanings:The difference of means was obtained by subtracting the mean of the five sample set from the mean of the 30 sample set, the standard deviation is the square root of the quantity of the sum of the squared deviations divided by 30 (since this is equal to 30 deviations). #$&* Analyze a set of 'made-up' time intervals and look at their distribution The set of numbers given below represents a set of 30 'made-up' quick-click time intervals. You will answer a few questions about this data set, including the mean and standard deviation of a 5-interval random sample. Later the results of all students will be compiled and used to demonstrate the 'sample standard deviation', which is an important statistical characteristic of sample and very relevant to interpretation of experimental results. .1752 .172 .1979 .1991 .176 .1711 .1664 .1665 .1858 .1764 .1765 .1885 .173 .1853 .1683 .1674 .1833 .1632 .1783 .1962 .1704 .1914 .1751 .1715 .1967 .1852 .1851 .1771 .1639 .1824 .1877 Copy these numbers into a cleared textbox, click on Mean and Standard Deviation, and report their mean and standard deviation in comma-delimited format in the first line below. Starting in the next line give a brief explanation of what your numbers mean and how you obtained them.your response &&&&&&&&&&&&&&&&&&
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mean and standard deviation of given data: 0.008771,0.01025 explanation and meanings:The mean deviation of the random sample is the sum of the differences divided by the numer of samples, in this case the sum divided by 31. The standard deviation is the square root of the quantity of the sum of the squared deviations divided by 31 (since this is more than 30 deviations). #$&* below, enter the following numbers, one to a line, in the given order: •The number which is two standard deviations less than the mean. •The number which is one standard deviation less than the mean. •The number which is equal to the mean. •The number which is one standard deviation more than the mean. •The number which is two standard deviations more than the mean. Starting in the next line give a brief explanation of what your numbers mean and how you obtained themyour response &&&&&&&&&&&&&&&&&&
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two standard deviations less than the mean:0.1586 one standard deviation less than the mean:0.16885 the mean:.1791 one standard deviation more than the mean:0.18935 two standard deviations more than the mean:0.1996 explanation and meanings: The standard deviation value was added and subtracted once or twice (to/from the mean) to find one or two standard devaitions from the mean. #$&* below, report each of the following numbers, one number to each line: •The number of the given time intervals which are less than the number which is two standard deviations less than the mean: •The number of the given time intervals which lie between two standard deviations less than the mean and one standard deviation less than the mean: •The number of the given time intervals which lie between one standard deviation less than the mean and the mean: •The number of the given time intervals which lie between the mean and one standard deviation more than the mean: •The number of the given time intervals which lie between one standard deviation more than the mean and two standard deviations more than the mean: •The number of the given time intervals which are greater than the number which is two standard deviations more than the mean: •Starting in the 7th line give a brief explanation of what your numbers mean and how you obtained them; as usual you may include optional comments:your response &&&&&&&&&&&&&&&&&&
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number of given intervals which are less than the number which is two standard deviations less than the mean:0 number of given intervals which are between two standard deviations less than the mean and one standard deviation less than the mean:6 number of given intervals which are between one standard deviation less than the mean and the mean:12 number of given intervals which are between the mean and one standard deviation more than the mean:8 number of given intervals which are between one standard deviation more than the mean and two standard deviations more than the mean:5 number of given intervals which are greater than the number which is two standard deviations more than the mean:0 explanation and comments: Most (20 of 31, 65%) values fell between plus/minus one standard devaition, and all values were within plus/minus two standard deviations. #$&* below, report each of the numbers you reported above, but expressed as a percent of the 30 intervals (rounded to the nearest percent). For example, the number 10 would be 33% of 30. Include a brief explanation of what your numbers mean and how you obtained themyour response &&&&&&&&&&&&&&&&&&
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percent of given intervals which are less than the number which is two standard deviations less than the mean:0 percent of given intervals which are between two standard deviations less than the mean and one standard deviation less than the mean:19% percent of given intervals which are between one standard deviation less than the mean and the mean:39% percent of given intervals which are between the mean and one standard deviation more than the mean:26% percent of given intervals which are between one standard deviation more than the mean and two standard deviations more than the mean:16% percent of given intervals which are greater than the number which is two standard deviations more than the mean:0 explanation and comments: To obtain the percentages i took each number in the answerr above as a percent of 31 (since there were 31 values listed). 65% of the values were within plus/minus one deviation, the remaining 35% were within plus minus two deviations. #$&* Perform a similar analysis with your 30-interval data Return to your own 30 time intervals. Count the numbers in each range (less than mean - 2 std dev, between mean - 2 std dev and mean - 1 std dev, between mean - 1 std dev and mean, etc.), using the mean and standard deviation of that data set. Report each number as a percent of your total number of intervals, one number in each of the first six lines below. Starting in the 7th line give a brief explanation of what your numbers mean and how you obtained themyour response &&&&&&&&&&&&&&&&&&
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number and percent of given intervals which are less than the number which is two standard deviations less than the mean: 0,0% number and percent of given intervals which are between two standard deviations less than the mean and one standard deviation less than the mean: 7,23% number and percent of given intervals which are between one standard deviation less than the mean and the mean: 7,23% number and percent of given intervals which are between the mean and one standard deviation more than the mean: 11,37% number and percent of given intervals which are between one standard deviation more than the mean and two standard deviations more than the mean: 4,13% number and percent of given intervals which are greater than the number which is two standard deviations more than the mean: 1,3% explanation and comments:I was suprised to see one value greater than 2 standard deviations, perhaps in a data set with large deviation this is not uncommon? #$&* In a standard 'normal' distribution, we expect that the respective percents in the six ranges will be about 2%, 14%, 34%, 34%, 14% and 2%. In a very large sample of data (say, at least tens of thousands of data points), if the data are in fact distributed normally, we expect actual results to very nearly reflect this distribution. If a large distribution does not closely match the expected results, we suspect that something in the system or in our observation process in fact deviates from the 'standard normal' expectation. Not everything we observe does in fact follow the standard normal pattern. You 3- interval results might or might not be expected to follow a standard normal distribution. If the data sample is not very large, the chance fluctuations in the distribution could have a significant effect on the percents, which in that case may not be all that close to the expected distribution. However in a medium-sized sample of 30 or so, we definitely expect more observations to lie in the middle two ranges and in either of the outer ranges, and we aren't too surprised if no results at all appear in the outermost ranges (more than 2 standard deviations from the mean). Based on the percents you reported and the percents quoted above, by how much would you say your actual 30-interval results deviated from the standard normal distribution? Did your results deviate enough to make you suspect that your clicks were not normally distributed about their mean?your response &&&&&&&&&&&&&&&&&&
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My results deviated from the standard normal distribution, if i had to quantify i would say it matched 75%, since 25 percentage points were outside the standard normal distribution, which may point to some button clicking habit, or may be just a factor of the data set. I would take a larger data set to determine if the button clicking was biased. #$&* Answer the same question for the 30 made-up time intervals given earlier.your response &&&&&&&&&&&&&&&&&&
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The made-up time intervals more closely matched the standard normal distribution, i would say an 87% match since only 13 percentage points were outside the standard normal distribution. #$&* Compare your distribution with the standard normal distribution We will in a subsequent exercise learn to sketch a standard normal curve, and to represent our information using this sketch. For the present, simply copy this figure below and label it as indicated below: There are five vertical lines on the graph, representing respectively •mean - 2 * std dev, also labeled z = -2 •mean - 1 * std dev, also labeled z = -1 •mean - std dev, also labeled z = 0 •mean + 1 * std dev, also labeled z = 1 •mean + 2 * std dev, also labeled z = 2. Label the x axis with the z numbers -2, -1, 0, 1 and 2. •Below these labels, place the respective numbers you obtained earlier for your 30-interval results, the numbers corresponding to mean - 2 * std dev, mean - 1 * std dev, etc.. •The five lines divide the region between the curve and the x axis into six smaller regions. Each of these regions will include either 2%, 14% or 34% of the total area between the curve and the x axis. •Within each region, write your best estimate of the number that represents its area as a percent of the total. below: •Indicate in the first line in comma-delimited format the numbers you placed in the regions, from left to right. •Indicate in the second line the x-axis labels corresponding to z = -2, -1, 0, 1 and 2 for the mean and standard deviation you obtained for your data. •Starting in the third line give a brief explanation of what your numbers mean and how you obtained them.your response &&&&&&&&&&&&&&&&&&
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I'm a bit confused about what to do here. I think i need to place the actual numbers i categorized into mean-2, mean -1, etc..? numbers representing percents in each of six regions: between two and one std dev less than mean: 0.984375,1.011719,1.015625,1.027344,1.027344,1.03125,1.03125 between one std dev less than mean and mean: 1.09375,1.046875,1.09375,1.058594,1.042969,1.0625,1.058594 between one std dev more than mean and mean: 1.152344,1.105469,1.136719,1.140625,1.140625,1.105469,1.140625,1.125,1.140625,1.136719,1.105469 between one and two std dev more than mean and mean: 1.21875,1.1875,1.183594,1.1875 greater than two std dev more than mean: 1.234375 x-axis labels for your data:-2, -1, 0, 1 and 2? as directed above, or am i confused? explanation:not sure- my bell curve looks is not symetrical like the ideal, but peaks between the mean and mean + 1 deviation #$&*