Critical Thinking and Experimental Design
Measures of Center and Variation and Relative Standing
Normal Distribution
Estimating a Population
Nico
100
During a criminal negligence lawsuit against a parachute manufacturer, the company brought forward evidence that every skydiver who responded to their survey found their chute totally effective. This is the problem with that argument.
What is missing data/volunteer bias? Solution: Dead skydivers can't complain.
100
The way wages are distributed (statistically) in America, as determined by the measures of center.
What is positively skewed? Solution: Thanks to the Warren Buffets of the world, mode<median<mean
100
This kind of distribution ends up resembling a Lego turned on it's side.
What is a uniform distribution? Solution: A uniform distribution, though it has an area of one just like standard normal distribution, is perfectly rectangular.
100
This equation corresponds to E, or the margin of error.
What is critical Z*sqrt((pq)/n)? Solution: That is how you find it, though it looks much better if you can write it out.
100
Because of politics, Nico was most curious about this when first starting Statistics.
What are confidence intervals? Solution: Throughout the 2016 election, I kept up with the polls, most which apparently called a Clinton victory overwhelmingly likely. This only exacerbated my shock when Trump pulled off his upset, but it did reveal just how little I understood about the 'margin of error' and 'levels of confidence' the pollsters were throwing around. I wanted to see where the analysis broke down, so I could avoid similar shocks in the future.
200
Though there are more sick people in the world than ever before, medicine can be said to be statistically successful on this basis.
What is per capita? Solution: An increase in everything is to be expected as a statistical population grows. What is important is whether such increases are proportionate.
200
The change in decimals for your answer as compared to your original value.
What is one more? Solution: Round-off rule.
200
These values put the standard in standard normal distribution.
What are population mean=0 and population standard deviation=1? Solution: Strictly speaking, all a normal distribution has to conform to is y=[e^-1/2*(x-mu/sigma)^2]/sigma*sqrt(2pi), or be 'bell-shaped' in simpler terms. It's only 'standard' if population mean and SD conform to those parameters.
200
To ensure that this value is always at least as large as it should be, it should always be rounded to the higher whole number. Thus, 20.03 would become 21.
What is sample size? Solution: The typical rules for rounding do not apply when trying to determine a sample size, because it's better to be safe than sorry and collect more data then less. At worst, you'll underestimate your confidence.
200
When doing math homework, Nico was pleasantly surprised and appreciated this curiosity within the textbook.
What is the writer's sense of humor? Solution: I always thought that math textbook's were as dry as you could get, but apparently statisticians are made of a different stock, because there were some surprisingly funny questions in this one. In particular, I enjoyed the one about conditional probability of getting a date after incorrectly guessing a women's age (0%). This made doing homework a lot more enjoyable, and helped me connect to statistics as a different sort of math.
300
I was technically saying this when I said "I'm unbeatable at darts. I increased my accuracy by 100% yesterday!"
What is I hit twice as much? Solution: When statistics are thrown around, it's important to keep track what 100% is. In this case, 100% meant 'of my original accuracy', but tried to imply 'in total'.
300
Thanks to this small change, we can turn standard deviation of a population into standard deviation of a sample.
What is adding -1 to the denominator? Solution: That's pretty much it. Works out on average.
300
Assuming that the time it takes for students to cram for portfolios is normally distributed, with a mean of 12 hours and a standard deviation of 4, this is the probability that a student crammed for at least 7 hours.
What is 89.44%? Solution: z=(7-12)/4 z=-1.25 or .1056 P(1-.1056=.8944)=89.44%
300
This method would be the one used given that the population standard deviation is unknown, but the population is normally distributed and the sample is a simple random sample.
What is the t distribution method? Solution: The normal distribution z method is nice and all, but it ultimately requires you already know standard deviation, which is rarely the case. The t distribution method has no such limitation.
300
Nico found himself surprisingly struggling with these ubiquitous little markers.
What are the symbols for all the variables? Solution: There were too many of them, half of them were latin symbols that you couldn't type or remember the name of, the rest were just letters with slight variation on the markings above, and some of them were barely used. For instance, what's the point of sample variance (s^2) when 99% of the time we use standard deviation? Why use lower case sigma for standard deviation and upper case sigma for summing the set? I could go on, but I just found it all very frustrating.
400
Line-Square-cube law is why these things tend to be misleading on graphs.
What are pictographs? Solution: A linear increase in dimension causes an exponential increase in area and volume.
400
DAILY DOUBLE At bare minimum, at least 93.75% of values lie within this range of the mean, according to Chebyshev's theorem.
What is within 4 standard deviations? Solution: Chebyshev gives us 1-1/k^2 1-1/4^2 1-1/16 15/16=93.75%
400
This width separates the top 20% of adult moose (mean of 103 centimeters, standard deviation of 2) from the bottom 80%.
What is 104.68 centimeters? Solution: Table look-up to find corresponding z-score for 80% gives us z=.084. 103+(0.84*2)=104.68 centimeters.
400
This is a 95% confidence interval estimate of the proportion of people who enjoy Vanilla to chocolate, given that our sample found that 152 people out of 580 preferred vanilla.
What is 0.226 < p < 0.298? Solution:95% corresponds to z=1.96^2. E=(1.96^2)*sqrt(.737931*0.2620289/580) E=+-.36.
400
Nico found this concept the most mathematically impressive of all the things covered this semester.
What are graphical distributions? Solution: Initially, I found the idea of converting statistical distribution into area in a graph confusing, but in hindsight it's really cool. There's an elegance to the idea of simply chopping up a bell and the corresponding area being equivalent to the proportion. I liked it.
500
Were it not for this fact, I'd believe the president when he said "90% of the UN Security Council is for this motion".
What is the number of members on the UN security council? Solution: 15 members. Unless 3/2 of a country is against the motion, that's not how it works.
500
The minimum speed a cat would need to be considered 'unusually' fast, given mean cat speed = 387 meters per hour, and sample variance= 25.
What is 397 meters per hour? Solution: Sample variance = standard deviation squared, ergo standard deviation = 5 meters per hour. A z score of + or - 2 is considered unusual, so 387+5*2=397.
500
This distance is the standard distribution of a group of exoplanets, which have a standard distribution centered around Earth and the middle 133 out of 140 were deemed 'earthlike' for falling within 500 kilometers of Earth's diameter (the mean).
What is 255.1 kilometers? 133/140=.95. So the middle 95% fell within 500 kilometers of Earth. This means we can deduce backwards our standard distribution, because by finding the z score that would get us 95% (97.5% on both sides), we find z= + - 1.96. Ergo, 500=1.96x 500/1.96=255.102=x
500
The % level (90% 95%, 99%) of a confidence interval refers to this, and not to the actual probability p falls within the interval.
What is the success rate of the process? Solution: This is a key but vital point, the consequence of living in a deterministic universe we use imperfect math to model instead of one where probabilistic estimations hold basis in reality; The confidence interval shows that given the construction of an arbitrary number of samples of size n, 95% of those confidence intervals would hit p.
500
Nico was most disappointed in himself with this aspect of classwork, and has voewd to change for the better on in next semester.
What is sticking to the formative work schedule? Solution: When we began the year in statistics, we came up with what I thought was a great system where I would turn in all my formative work for the chapter with the test. It simplified things for everyone involved, and for a time I actually adhered to it. However, one thing led to another and I fell into my old habits again. I'm personally rather ashamed of this, and wish I had stuck to the system we had instead of cramming assignments in whenever it was convenient. I promise to do better next semester.
M
e
n
u