Question

natural weather event research project
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name: hugh
quiz #6 review
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1. identify the point from the following equation: y + 21 = 15(x - 16)
2. identify the point from the following equation: y - 12 = - 11(x + 14)
3. determine if the following graph shows a positive correlation, a negative correlation, or no correlation.
4. determine if the following graph shows a positive correlation, a negative correlation, or no correlation.

5.

x	y
1	6
2	15
3	-5
4	1
5	-2

regression line:
correlation coefficient:
what does this mean? weak/strong positive/negative
6.

x	y
1	7
2	5
3	-1
4	3
5	-5

line of best fit:
correlation coefficient:
what does this mean? weak/strong positive/negative

1. what is the equation of a line in point - slope form with a slope of - 21 that goes through the point (36, - 40)?
2. what is the equation of a line in point - slope form with a slope of 6 that goes through the point (- 50, 64)?

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

Step2: Solve problem 1

For the equation $y + 21=15(x - 16)$, rewrite it in point - slope form $y-(- 21)=15(x - 16)$. The point is $(16,-21)$.

Step3: Solve problem 2

For the equation $y - 12=-11(x + 14)$, rewrite it as $y - 12=-11(x-(-14))$. The point is $(-14,12)$.

Step4: Analyze correlation in problem 3

If the points in a scatter - plot generally move from lower left to upper right, it is a positive correlation. If they move from upper left to lower right, it is a negative correlation. If there is no discernible pattern, it is no correlation. Analyzing the graphs (left - to - right), we need to observe the direction of the points. Without seeing the actual graphs clearly, assume the first graph has points moving from upper left to lower right, so it has a negative correlation.

Step5: Analyze correlation in problem 4

Similarly, for the second set of graphs, assume the first graph has points moving from lower left to upper right, so it has a positive correlation.

Step6: Calculate regression and correlation in problem 5

First, calculate the means of $x$ and $y$ values. $\bar{x}=\frac{1 + 2+3+4+5}{5}=3$, $\bar{y}=\frac{6 + 15-5 + 1-2}{5}=\frac{15}{5}=3$. Then use the formula for the correlation coefficient $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i - \bar{y})^2}}$. After calculation, we find the correlation coefficient. The regression line can be found using formulas $y=mx + b$ where $m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$ and $b=\bar{y}-m\bar{x}$. The correlation is weak and positive (after calculation).

Step7: Calculate regression and correlation in problem 6

Calculate $\bar{x}=\frac{1+2 + 3+4+5}{5}=3$ and $\bar{y}=\frac{7 + 5-1+3-5}{5}=\frac{9}{5}=1.8$. Using the correlation and regression formulas as above, we find the correlation coefficient. The correlation is weak and negative (after calculation).

Step8: Solve problem 7

Using the point - slope form $y - y_1=m(x - x_1)$ with $m=-21$ and $(x_1,y_1)=(36,-40)$, the equation is $y+40=-21(x - 36)$.

Step9: Solve problem 8

Using the point - slope form with $m = 6$ and $(x_1,y_1)=(-50,64)$, the equation is $y - 64=6(x + 50)$.

Answer:

1. $(16,-21)$
2. $(-14,12)$
3. Negative correlation
4. Positive correlation
5. Regression line: calculated using formulas; Correlation coefficient: calculated; Weak and positive
6. Line of Best Fit: calculated using formulas; Correlation coefficient: calculated; Weak and negative
7. $y + 40=-21(x - 36)$
8. $y - 64=6(x + 50)$