Lies, Damned Lies, or Statistics- How to Tell the Truth with Statistics, 2017a

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38 2. BI-VARIATE STATISTICS: BASICS Exercises EXERCISE 2.1. Suppose you pick 50 random adults across the United States in January 2017 and measure how tall they are. For each of them, you also get accurate information about how tall their (biological) parents are. Now, using as your individuals these 50 adults and as the two variables their heights and the average of their parents’ heights, make a sketch of what you think the resulting scatterplot would look like. Explain why you made the choice you did of one variable to be the explanatory and the other the response variable. Tell what are the shape, strength, and direction you see in this scatterplot, if it shows a deterministic or non-deterministic association, and why you think those conclusions would be true if you were to do this exercise with real data. Is there any time or place other than right now in the United States where you think the data you would collect as above would result in a scatterplot that would look fairly different in some significant way? Explain! EXERCISE 2.2. It actually turns out that it is not true that the more a person works, the more they produce ... at least not always. Data on workers in a wide variety of industries show that working more hours produces more of that business’s product for a while, but then after too many hours of work, keeping on working makes for almost no additional production. Describe how you might collect data to investigate this relationship, by telling what individuals, population, sample, and variables you would use. Then, assuming the truth of the above statement about what other research in this area has found, make an example of a scatterplot that you think might result from your suggested data collection. EXERCISE 2.3. Make a scatterplot of the dataset consisting of the following pairs of measurements: {(8, 16), (9, 9), (10, 4), (11, 1), (12, 0), (13, 1), (14, 4), (15, 9), (16, 16)}. You can do this quite easily by hand (there are only nine points!). Feel free to use an electronic device to make the plot for you, if you have one you know how to use, but copy the resulting picture into the homework you hand in, either by hand or cut-and-paste into an electronic version. Describe the scatterplot, telling what are the shape, strength, and direction. What do you think would be the correlation coefficient of this dataset? As always, explain all of your reasoning!
CHAPTER 3 Linear Regression Quick review of equations for lines: Recall the equation of a line is usually in the form y = mx + b, wherex and y are variables and m and b are numbers. Some basic facts about lines: • If you are given a number for x, you can plug it in to the equation y = mx + b to get a number for y, which together give you a point with coordinates (x, y) that is on the line. • m is the slope, which tells how much the line goes up (increasing y) for every unit you move over to the right (increasing x) – we often say that the value of the slope is m = rise.Itcanbe run – positive, if the line is tilted up, – negative, if the line is tilted down, – zero, if the line is horizontal, and – undefined, if the line is vertical. • You can calculate the slope by finding the coordinates (x 1 ,y 1 ) and (x 2 ,y 2 ) of any two points on the line and then m = y 2−y 1 x 2 −x 1 . • In particular, x 2 − x 1 =1,thenm = y 2−y 1 = y 1 2 − y 1 – so if you look at how much the line goes up in each step of one unit to the right, that number will be the slope m (and if it goes down, the slope m will simply be negative). In other words, the slope answers the question “for each step to the right, how much does the line increase (or decrease)?” • b is the y-intercept, which tells the y-coordinate of the point where the line crosses the y-axis. Another way of saying that is that b is the y value of the line when the x is 0. 3.1. The Least Squares Regression Line Suppose we have some bivariate quantitative data {(x 1 ,y 1 ),...,(x n ,y n )} for which the correlation coefficient indicates some linear association. It is natural to want to write down explicitly the equation of the best line through the data – the question is what is this line. The most common meaning given to best in this search for the line is the line whose total square error is the smallest possible. We make this notion precise in two steps 39
Page 1: Lies, Damned Lies, or Statistics: H
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Page 12 and 13: The first instinct of the scientist
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Part 3 Inferential Statistics
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CHAPTER 6 Basic Inferences The purp
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6.1. THE CENTRAL LIMIT THEOREM 113
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6.2. BASIC CONFIDENCE INTERVALS 115
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EXERCISES 125 Exercises EXERCISE 6.
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Bibliography [Gal17] Gallup, Presid
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130 INDEX causation, 46 center of a
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