Overview. All of the material in this supplemental online text is purely optional and is meant to enhance the textbook. Use whatever aspects of it help to deepen your understanding of the material and ignore the rest. It assumes you have already read the relevant chapter. I will be adding more material as the semester progresses.
In teaching about statistics--the primary tool scientists use to make sense of their data,--it is important to me that I elucidate the differences between the scientific approach to knowledge and the religious approach to knowledge. It is also important to me that I point out that both approaches have value, depending upon the context in which they are used. I would like to take advantage of this opportunity of creating supplemental information for the text, to expand this topic to the point where it encompasses a topic that is important to me and that I would like to share. This involves talking at a conceptual level that is far beyond where the rest of this supplemental material will reside. After this chapter I will settle down to simply giving additional examples of the story problems covered in later chapters of the text.
In the first chapter of the text I describe the fundamental attributes of the scientific approach to knowledge. These attributes make the scientific approach distinctly different from other approaches, including religion. What I would like to add to that description is that while the scientific approach and the religious approach have important, fundamental, differences, they both arose from the same culture and at a deep level they share some basic assumptions about the nature of reality. These assumptions are rarely brought to light to be examined because, obviously, they are assumed to be true.
The dictionary defines a worldview as a culture's set of concepts and beliefs about the nature of reality. Both science and Western religion reside within the modern Western worldview. There are other worldviews on the planet, and there have been other worldviews in the Western world in earlier times. If we know only one worldview we tend to think it is the only one that exists and that all other worldviews are simply variations of our own. In the case of indigenous worldviews we tend to think of them as primitive versions of our Western worldview, like our own worldview but with more superstitions and less knowledge.
For the past 24 years I have been immersing myself in the worldview of the indigenous people who live in the high Andes of Peru. Their worldview, the Andean Cosmovision, supports a way of perceiving and interacting with reality that is fundamentally different than that of the Western worldview. It is a way of understanding reality that has not been influenced by the Bible, or by the classic Greek philosophers, or by Descartes' division of reality into separate mind and matter, or by the scientific revolution. This worldview simply cannot be understood through the spectacles of science or Western religion, which we in the West assume can handle anything. I speak of the Andean Cosmovision simply because it is the only worldview which which I am intimately familiar other than my own modern, Western, worldview, but I believe the same holds true for other worldviews as well.
So my point is this. In the textbook I have described the important differences between science and Western religion. They are two distinct approaches to knowledge. They both arose, however, from the same worldview and share some basic assumptions about the nature of reality. When we step out of the Western worldview, and into other worldviews we can find other ways of perceiving and interacting with reality that cannot be encompassed, described, or understood from within the Western worldview. If we don't recognize this, then when we consider other worldviews, rather than looking through a window at a new way of perceiving reality we are instead simply looking into a mirror.
If you would like to know more about the Andean Cosmovision you can visit my blog at Salka Wind Blog where I have published a great deal of information stemming from my research in Peru.
Identify each of the following measurement scales. Click "See Answer" to see answer (did I need to say that?).
This is a cardinal measure as it directly measures a quantity (number of students).
This is a nominal scale as the numbers reflect qualitative (categorical) differences rather than quantities.
This is an ordinal scale. The scores reflect a change in quantity in a specific direction (the greater the score the more likely to purchase a phone). The sizes of the steps are not necessarily equal (i.e. the difference between 'unlikely' and 'somewhat likely' may not be the same size as the difference between 'somewhat likely' and 'very likely').
This is a rank scale. While the size of a university is a cardinal scale (see above) what is being measured here is how the university ranks in comparison to other universities. The score for the largest university would be "1" which tells us how it compares to others but does not actually tell us how many students there are. While rank scales are a subset of ordinal scales, what makes a rank scale different is that the score is dependent upon how the subject compares to others in some group. While the largest university in Utah would receive a rank score of "1" if we compared it to others in the Utah, it would get a different rank score if we compared it to other universities in the whole country.
This is an ordinal scale. The scores reflect a change in quantity in a specific direction (the greater the score the lower the level of satisfaction) and the sizes of the steps are not necessarily equal.
This is a rank scale. Your score reflects your ranking in birth order in your family.
This is a nominal scale.
This is a cardinal scale.
You sample from a population and obtain the following scores. Y = 8, 7, 5, 7, 6, 4
Descriptive Statistics. Compute the following, be sure to use the correct symbols.
Inferential Statistics. Compute the following, be sure to use the correct symbols.
Eyeballing the standard deviation.
Finding proportions under the normal curve.
You have a population that is normally distributed with a mean of 80 (i.e. μ = 80) and a standard deviation of 7 (i.e. σ = 7). What proportion of the scores will be 76 or greater (i.e. Y ≥ 76)?
Draw a normal curve, label it "Original Population (Y Scores)", mark the mean and standard deviation on the curve, mark (approximately) where Y=71 and Y=89 would fall, and then shade in the areas in question (the areas where Y is less than or equal to 71 and Y is greater than or equal to 89).
The only way to determine the proportion of scores that fall in the shaded areas is by changing the Y values to z values so that we can use the Normal Tool. The formula below is used to change Y=71 and Y=89 into z scores.
Those z values have been place on the curve below.
Now use the Normal Tool to find the proportion of the curve that falls in the shaded areas.
The Normal Tool tells us that 0.5485 proportion of the scores fall in the shaded areas, which is our answer.
Note: When we want to determine what proportion of the sample means fall within certain values, we turn to the sampling distribution of the mean (SDM).
Determine the sampling distribution of the mean (SDM).
Draw a normal curve, label it "SDM for N=10", mark the mean and standard deviation on the curve. This is a graph of sample means, mark (approximately) where M=71 and M=89 would fall, and then shade in the areas in question (the areas where M is less than or equal to 71 and M is greater than or equal to 89).
The only way to determine the proportion of means that fall in the shaded areas is by changing the M values to z values so that we can use the Normal Tool. The formula below is used to change M=71 and M=89 into z scores. Note that what has changed from the computation of z above is that the denominator is different as the standard deviation of the curve is different.
We know have the following curve and we can again use the Normal Tool to find what proportion of the curve is in the shaded regions. The only thing that has changed in using the Normal Tool is that we now have different values for z.
The Normal Tool tells us that .0588 proportion of the sample means will be 71 or less or 89 or greater (i.e. 8 or more away from the population mean).
You sample six scores from a population and obtain the following scores. Y = 115, 120, 109, 120, 118, 106.
Compute the 95% confidence interval of the mean.
N = 6
Now we need to know the t value that cuts off 5% of the curve (2.5% on each tail) given our degrees of freedom (df).
df = N - 1 = 5
Go to the t Tool in Oak Software.
The t Tool screen should look like this:
When you press "Calculate" you should get t ± 2.5705.
We can now calculate the 95% confidence interval:
So the 95% confidence interval is: 108.45 ≤ μ ≤ 120.89.
There are two, acceptable, ways to interpret this:
Story problem. You are testing a theory which predicts that Population 1 should have higher scores than Population 2. That you are specifically predicting which population should have the higher scores makes this a directional hypothesis that should be analyzed with a one-tailed test. We will skip all of the number crunching and just look at which tail you shade (which will influence the p value you get).
Write Ha to reflect the prediction: μ1 > μ2
Write H0 to cover everything else: μ1 ≤ μ2
Mark the approximate location of t on the curve.
Now the question becomes do we shade in the area to the left of 't' or the area to the right of 't'? To answer that we need to look at what Ha predicts. Ha says that μ1 is greater than μ2, if that is true then M1 should be greater than M2. If we look at the formula for 't' we can see that whether 't' is negative or positive is determined by whether M1-M2 is negative or positive. Since Ha predicts that M1 should be greater than M2, then if Ha is true M1-M2 should be a positive number, and thus 't' will be positive as well. On the curve we shade in the area where Ha says the results should fall. Ha says that 't' should be positive, which is to the right on the curve, so we shade in that area. The 'idiot-proof' approach is to look at Ha and treat the '>' as an arrow pointing to the tail to shade. We now know which area to select on the Oak Software tool.
Story Problem: You have a theory which predicts that Population 1 should have lower scores than Population 2, which makes this a one-tailed test.
Ha: μ1 < μ2
Analysis: the two-tailed p value is .08
To find the one-tailed p value from this two-tailed p value you need to look at the sample means to see if they fit the theory's prediction. If the theory is correct then M1 should be less than M2 (see Ha above). Let's say that when we look at the sample means we see that M1=34 and M2=56. That fits the theory's prediction!
Now let's say that when we look at the sample means we see that M1=78 and M2=65. The theory's prediction was wrong!