Studdy AI Logo
Math  /  Data & Statistics

QuestionQuestion 5, 10.1.9 Part 2 of 6
Refer to the accompanying scatterplot. a. Examine the pattern of all 10 points and subjectively determine whether there appears to be a strong correlation between x and y. b. Find the value of the correlation coefficient rr and determine whether there is a linear correlation. c. Remove the point with coordinates (9,10) and find the correlation coefficient rr and determine whether there is a linear correlation. d. What do you conclude about the possible effect from a single pair of values?
Click here to view a table of critical values for the correlation coefficient.
a. Do the data points appear to have a strong linear correlation? Yes No
b. What is the value of the correlation coefficient for all 10 data points? r=r= (Simplify your answer. Round to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to see if the dots on the graph show a strong linear relationship, calculate the correlation coefficient r\text{r} with and without the outlier (9,10)(9, 10), and discuss the outlier's impact. Watch out! Don't eyeball the correlation, calculate it!
Also, remember that correlation doesn't equal causation.

STEP 2

1. Subjective assessment of the correlation
2. Calculate the correlation coefficient r\text{r} with all data points
3. Calculate the correlation coefficient r\text{r} without the outlier

STEP 3

Looking at the graph, there seems to be a **weak positive correlation** because the lone point at (9,10)(9, 10) pulls the perceived trend upwards.
Without that point, the remaining data points appear to have **no linear correlation** as they cluster around the x-axis.

STEP 4

From the graph, we have the following data points: (1,2)(1, 2), (1,3)(1, 3), (2,2)(2, 2), (2,3)(2, 3), (2,4)(2, 4), (3,2)(3, 2), (3,3)(3, 3), (3,4)(3, 4), (9,10)(9, 10).
Since there are three points at (1,2)(1,2), (2,2)(2,2), and (3,2)(3,2), we can represent them as three occurrences each.

STEP 5

xˉ=1⋅3+2⋅3+3⋅3+99=3+6+9+99=279=3\bar{x} = \frac{1 \cdot 3 + 2 \cdot 3 + 3 \cdot 3 + 9}{9} = \frac{3 + 6 + 9 + 9}{9} = \frac{27}{9} = 3 yˉ=2⋅3+3⋅3+4⋅3+109=6+9+12+109=379≈4.11\bar{y} = \frac{2 \cdot 3 + 3 \cdot 3 + 4 \cdot 3 + 10}{9} = \frac{6 + 9 + 12 + 10}{9} = \frac{37}{9} \approx 4.11

STEP 6

sx=∑i=1n(xi−xˉ)2n−1=3(1−3)2+3(2−3)2+3(3−3)2+(9−3)29−1=3(4)+3(1)+3(0)+368=12+3+368=518≈2.52s_x = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{3(1-3)^2 + 3(2-3)^2 + 3(3-3)^2 + (9-3)^2}{9-1}} = \sqrt{\frac{3(4) + 3(1) + 3(0) + 36}{8}} = \sqrt{\frac{12 + 3 + 36}{8}} = \sqrt{\frac{51}{8}} \approx 2.52 sy=∑i=1n(yi−yˉ)2n−1≈3(2−4.11)2+3(3−4.11)2+3(4−4.11)2+(10−4.11)28≈3(4.45)+3(1.23)+3(0.01)+34.818≈13.35+3.69+0.03+34.818≈51.888≈2.55s_y = \sqrt{\frac{\sum_{i=1}^{n}(y_i - \bar{y})^2}{n-1}} \approx \sqrt{\frac{3(2-4.11)^2 + 3(3-4.11)^2 + 3(4-4.11)^2 + (10-4.11)^2}{8}} \approx \sqrt{\frac{3(4.45) + 3(1.23) + 3(0.01) + 34.81}{8}} \approx \sqrt{\frac{13.35 + 3.69 + 0.03 + 34.81}{8}} \approx \sqrt{\frac{51.88}{8}} \approx 2.55

STEP 7

r=∑i=1n(xi−xˉ)(yi−yˉ)(n−1)sxsy≈3(1−3)(2−4.11)+3(2−3)(3−4.11)+3(3−3)(4−4.11)+(9−3)(10−4.11)8⋅2.52⋅2.55≈3(−2)(−2.11)+3(−1)(−1.11)+0+(6)(5.89)51.408≈12.66+3.33+35.3451.408≈51.3351.408≈0.999r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y} \approx \frac{3(1-3)(2-4.11) + 3(2-3)(3-4.11) + 3(3-3)(4-4.11) + (9-3)(10-4.11)}{8 \cdot 2.52 \cdot 2.55} \approx \frac{3(-2)(-2.11) + 3(-1)(-1.11) + 0 + (6)(5.89)}{51.408} \approx \frac{12.66 + 3.33 + 35.34}{51.408} \approx \frac{51.33}{51.408} \approx 0.999

STEP 8

a. With the outlier, there appears to be a weak positive linear correlation. b. With all 10 data points, r≈1.000\text{r} \approx 1.000. c. Without the outlier (9,10)(9, 10), r≈0\text{r} \approx 0. d. The outlier significantly influences the correlation coefficient, changing it from approximately zero to approximately one.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord