(02.01 mc)
a bread distributor conducts regular surveys of consumer spending on its products. the average weekly household spending (in dollars) on packaged bread and bakery bread for each of 12 states was recorded. a scatterplot of spending on packaged bread versus bakery bread is shown. which of the following statements is true?
scatterplot image
options:
- there is evidence of no association between spending on bakery bread and packaged bread.
- the correlation for these data is r = -0.5.
- point (7, 2) is an outlier.
- the equation of the least - squares regression line for this plot is approximately ŷ = 8 - 7x.
- there are no overly influential observations that affect the least - squares regression line.

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Accepted Answer

# Brief Explanations:
1. Analyze the first option: The scatterplot shows a negative trend (as packaged bread spending increases, bakery bread spending tends to decrease), so there is an association. Eliminate this option.
2. Analyze the second option: The correlation $ r = -0.5 $ suggests a moderate negative linear relationship. Looking at the scatterplot, the points show a negative trend, and $ r=-0.5 $ is a reasonable estimate for the correlation (not too strong, not too weak, negative).
3. Analyze the third option: Point (7, 2) - looking at the scatterplot, most points with packaged bread around 7 have bakery bread around 10 or so, but (7,2) is much lower? Wait, no, let's check the axes. Wait, the x - axis is packaged bread (0 - 8), y - axis is bakery bread (0 - 14). Wait, when packaged bread is 7, there are points at (7,10) and (7,2). Wait, but is (7,2) an outlier? Wait, no, let's see the general trend. Wait, actually, the other points with packaged bread 7: one is at 10, one at 2. But maybe not. Wait, let's re - evaluate. Wait, the fourth option: Are there overly influential observations? Let's see, the point (1,12) - if we remove it, would the regression line change a lot? Maybe, but the second option: the correlation. Wait, the scatterplot has a negative linear trend. The correlation $ r=-0.5 $ is a possible value. Wait, maybe I made a mistake. Wait, no, let's check again.

Wait, the first option: there is an association (negative), so eliminate. The third option: Point (7,2) - is it an outlier? Let's see the y - value for x = 7. Most points at x = 7 have y around 10, but (7,2) is low. But maybe not. Wait, the equation $ \hat{y}=8 - 7x $: the slope is - 7, which would mean a very strong negative relationship, but the scatterplot doesn't show a very strong relationship. So the equation is wrong. The fifth option: Are there no overly influential observations? Maybe, but the second option: $ r=-0.5 $ is a moderate negative correlation, which matches the scatterplot's negative trend (not too strong, not too weak). Wait, but let's check the third option again. Wait, the point (7,2): when x = 7, the other point is (7,10), so (7,2) is far from the other point at x = 7, but is it an outlier? Maybe, but the second option: the correlation. Wait, maybe I was wrong. Wait, no, let's think about correlation. The scatterplot has a negative linear association. The correlation coefficient $ r $ ranges from - 1 to 1. A value of $ r=-0.5 $ indicates a moderate negative linear relationship, which is consistent with the scatterplot (the points don't lie perfectly on a line, but have a negative trend). The point (7,2): is it an outlier? Let's see the y - value for x = 7. The other point at x = 7 is (7,10), so (7,2) is an outlier? But maybe not. Wait, the equation $ \hat{y}=8 - 7x $: if x = 1, $ \hat{y}=1 $, but the point (1,12) is far from that, so the equation is wrong. The first option: there is an association, so wrong. The fifth option: Are there no overly influential observations? Maybe, but the second option: $ r=-0.5 $ is a reasonable correlation for this scatterplot.

Wait, no, I think I made a mistake. Let's re - examine the options.

Option 1: There is an association (negative), so false.

Option 2: Correlation $ r = - 0.5 $. The scatterplot shows a negative linear trend, and $ r=-0.5 $ is a moderate negative correlation, which is plausible.

Option 3: Point (7,2) - is it an outlier? Let's see the y - value for x = 7. The other point at x = 7 is (7,10), so (7,2) is an outlier? But maybe not. Wait, the y - axis for x = 7: most points are around 10, but (7,2) is low. But maybe the answer is Option 2. Wait, no, wait the third option: Point (7,2) - when x = 7, the expected y (from the trend) would be around, say, if the correlation is - 0.5, the trend is negative. But (7,2) is an outlier? Maybe. But I think the correct answer is Option 2: The correlation for these data is $ r=-0.5 $.

Wait, no, maybe I messed up. Let's check the axes again. The x - axis is packaged bread (0 - 8), y - axis is bakery bread (0 - 14). The points:

(1,12), (2,6), (3,2), (3,11), (4,7), (5,8), (5,6), (6,5), (6,10), (7,10), (7,2), (1,12)? Wait, no, the scatterplot:

x = 1: (1,12)

x = 2: (2,6)

x = 3: (3,2), (3,11)

x = 4: (4,7)

x = 5: (5,8), (5,6)

x = 6: (6,5), (6,10)

x = 7: (7,10), (7,2)

So, the general trend: as x (packaged bread) increases, y (bakery bread) tends to decrease. So negative correlation. The correlation $ r=-0.5 $ is a moderate negative correlation, which fits. The point (7,2): is it an outlier? Among the points with x = 7, one is at y = 10, one at y = 2. So (7,2) is an outlier? But the second option: $ r=-0.5 $ is a possible correlation. Wait, maybe the answer is Option 2.

# Answer:
B. The correlation for these data is $ r=-0.5 $

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