ritten Problems 1. Question 3.49 The figure shows recent data on x the number of
ID: 3353659 • Letter: R
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ritten Problems 1. Question 3.49 The figure shows recent data on x the number of televisions per 100 people and y-the birth rate (number of births per 1,000 people) for six African and Asian countries. The regression line, 9-29.8-0.024x, applies to the data for these six countries. For illustration, another point is added at (81,15.2), which is the observation for the United States. The regression line for all seven points is y-31.2-0.195x. The figure shows this line and the one without the U.S. observation and Number of TVs per 100 People 35 United Sates 30 25 with United State 20 15 20 40 60 Number of TVs a. Does the U.S. observation appear to be () an outlier on x, (i) an outlier on y, or (ii) a b. State the two conditions under which a single point can have a dramatic effect on c. This one point also drastically affects the correlation, which is r-0.051 without the regression outlier relative to the regression line for the other six points? the slope and show that they apply here. United States but r=-0.935 with the United States. Explain why you would conclude that the association between birth rate and number of televisions is (i) very weak without the U.S. point and (i) very strong with the U.S. point. d. Explain why the U.S. residual for the line fitted using that point is very small. This shows that a point can be influential even if its residual is not large.Explanation / Answer
a. The U.S. observations is a regression outlier relative to the regression line for the other six points. This point is heavily contorted out of both the x and y bunching of other 6 countries. That is, it has a much higher rate of televisions per 100 people, and a much lower birth rate. So in both respects, it does not fit with the other 6 data, which are pretty fairly bunched together on an approximate line
b. One way that a point can have a dramatic effect on slope, as it applies here, is that it has the opposite tendency (rising or declining). For example, here the points are showing a slight negatuve correlation, whereas the isloated U.S. point gives a much more pronounced negative drop whrereby for a higher x, the y value is drastically reduced relative to the y value of other points.
Another way to effect slope is that if the point is almost vertically above or vertically below one of the data points in the initial set. That is, for the same x value, the new data point has a much different y value.
c. A correlation is generally considered very weak if r lies between -0.2 and 0.2. Without the U.S. point, the value of r is very close to 0 viz. -0.051. So it shows very weak correlation. It is evident because the regression line has points widely lying on either side of the regression line. With the U.S. point, the regression coefficient is very close to -1, viz. -0.935. We generally consider strong correlarion when r is less than -0.8 or more than 0.8. As can also be seen in the curve, the points are now much more closely bunched on the regression line.
d. As ic clearly evident, the regression line using U.S. point almost exactly passes through the point itself. In other words, the U.S. point seems to be a "trednsetter" in the sense that it redefines the slope such that all the points are a much better fit on the new line. This primarily becomes possible because the x value of U.S. point is widely separated from that of all other points, hence giving a much more pronounced effect from its distinct y value.
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