Statistics Worksheet 1 - Solutions

Transcription

1 Statistics Worksheet 1 - Solutions Math& 146 Descriptive Statistics (Chapter 2) Data Set 1 We look at the following data set, describing hypothetical observations of voltage of as et of 9V batteries. The data is sorted to make it easier to identify some indexes We also have the following summaries: count 32 sum sum of squares For this data compute ( by hand, that is, not using a spreadsheet, but calculators are fine) 1. The mean 2. The population variance and population standard deviation 3. The sample variance and sample standard deviation 4. The median 5. The first quartile 6. The third quartile 7. The minimum 8. The maximum 9. The range 10. The midrange Don't draw any conclusion, since we are told nothing about how this data was collected.

2 Solutions Here are the answers provided by the same program that generated the data: Mean quartile Median quartile Population Standard Deviation Population Variance Sample Standard Deviation Sample Variance Range Minimum Maximum Midrange You will notice that the quartiles do not correspond to the midpoints between the 8 th and 9 th, and between the 24 th and 25 th data point. The program uses one of the fancier choices, but fact is that any number between the two qualifies as a quartile. The same applies to all other examples/ The data was simulated assuming what is called a Normal model (capitalized, since Normal here is a technical term, not a synonym for run-of-the-mill). In theory, a very large sample should produce a histogram closely resembling the following graph:

3 Here are two histograms, with different bins, of the actual data we are looking at:

4 Data Set 2 We look at the following data set, describing very hypothetical observations of interarrival times of a bus at a bus stop. The data is sorted to make it easier to identify some indexes We also have the following summaries: Count 40 Sum Sum of squares For this data compute ( by hand, that is, not using a spreadsheet, but calculators are fine) 1. The mean 2. The population variance and population standard deviation 3. The sample variance and sample standard deviation 4. The median 5. The first quartile 6. The third quartile 7. The minimum 8. The maximum 9. The range 10. The midrange Don't draw any conclusion, since we are told nothing about how this data was collected.

5 Solutions Here are the answers provided by the same program that generated the data: Mean quartile Median quartile Population Standard Deviation Population Variance Sample Standard Deviation Sample Variance Range Minimum Maximum Midrange The data was simulated assuming what is called an Exponential model. This model fits interarrival times of events that are completely unconnected typically, requests for service to a common resource (like a printer) in a vast network, or, as its original application, phone calls arriving at a central city switchboard (at the time when these things existed) In theory, a very large sample should produce a histogram closely resembling the following graph: Notice how the tail (the left end of the graph) approaches 0 at a slower pace than the preceding ( Normal ) picture. Also the curve is not symmetric at all.

6 Here are two histograms, with different bins, of the actual data we are looking at:

7 Data Set 3 We look at the following data set, describing hypothetical observations of yearly income, in units of $10,000. The data is sorted to make it easier to identify some indexes We also have the following summaries: count 24 sum sum squares For this data compute ( by hand, that is, not using a spreadsheet, but calculators are fine) 1. The mean 2. The population variance and population standard deviation 3. The sample variance and sample standard deviation 4. The median 5. The first quartile 6. The third quartile 7. The minimum 8. The maximum 9. The range 10. The midrange Don't draw any conclusion, since we are told nothing about how this data was collected.

8 Solutions Here are the answers provided by the same program that generated the data: Mean quartile Median quartile Population Standard Deviation Population Variance Standard Deviation Sample Variance Range Minimum Maximum Midrange The data was simulated assuming what is called a Pareto model. This is one of a category of models that will produce widely scattered data, with heavy tails, that is the histograms should approach zero very slowly. These models have become popular after events like the financial crash of 2007 suggested that Normal models were inadequate for the description of markets. In theory, a very large sample should produce a histogram closely resembling the following graph: This is an example of power law, since the graph is that of an inverse power (here, proportional to x 3/ 2 ), which approaches 0 much slower than Normal (which looks like e x2 ), or Exponential (which looks like e x )

9 Here are two histograms, with different bins, of the actual data we are looking at: The computer drawn pictures do not quite render the situation: here are the tables listing the bins and their frequency: Table 1 Table 2 from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to 1 from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to below from to 1

10 As you can see, most of the data is clustered at the beginning, but there is one that is extremely larger than the others. If we trust the software to have done its job faithfully, we should not discard this supposed outlier. Following a popular book about unexpected outcomes, this could be called a Black Swan surprising, but a possible outcome, even if it could be disruptive of normality (in the colloquial sense, but, as it happens, also in the technical sense of the word).