Part B
The scatter plot has an upward trend going on. As the amount of money goes up, the happiness score goes up overall as well. More abstractly, we see that as x goes up, so does y (in general). We consider this a positive correlation. This is what you should get for your scatterplot
where,
x = annual salary in dollars
y = happiness score (on a scale of 0 to 100)
side note: if you computed the correlation coefficient r, then you should get roughly r = 0.81; this is a fairly strong positive correlation. The closer r gets to r = 1, the stronger the positive correlation gets.
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Part C
Plug x = 67000 into the regression line and simplify
y = 0.0006*x + 62
y = 0.0006*67000 + 62 ..... replace every x with 67000
y = 40.2 + 62
y = 100.2
Since the happeiness score can only be between 0 and 100, this means we should round down to 100. Therefore, if someone is making $67,000, then we estimate their happiness score to be a perfect 100. Of course, the true y value for x = 67000 is likely to be different from this result. Regression lines only have a limited useful application.
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Part D
The 0.0006 is the slope of the regression line. Recall that y = mx+b is the general slope intercept equation. So in this case, m = 0.0006. The slope represents the rate of change, or growth. It measures how fast y is increasing if we increase x the same amount each time. If we increase x by 1, then y goes up by 0.0006. This is the same as saying "each time we increase x by 10,000, then y goes up by 6". Note how I multiplied both parts by 10,000.
The interpretation of the slope could be: "for each additional $10,000 you make, your happiness score is estimated to increase by 6 points"
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Go back to the y = mx+b form and compare that to y = 0.0006x + 62. We see that b = 62 is the y intercept. So on the vertical number line, the graph crosses 62.
The location of this point is (0,62)
Proof of this can be seen if we plugged in x = 0
y = 0.0006*x + 62
y = 0.0006*0 + 62 ..... replace every x with 0
y = 0 + 62
y = 62
The x = 0 basically eliminates the whole term 0.0006*x leaving behind just the 62
The interpretation of the y intercept is: "if you make 0 dollars, then your happiness score is estimated to be 62". This is the floor or lowest value for the happiness score. This is because there's an upward trend, meaning the happiness score (y) will not go down as the salary amounts (x) increase. As stated earlier, this is an estimate of the y values and not always a true reflection of what y really is.