Question

Height vs Weight - Erroneous Data: You will need to use software to answer these questions.

Below is the scatterplot, regression line, and corresponding data for the height and weight of 11 randomly selected adults. You should notice something odd about the last entry.

index height (x) weight (y) inches pounds 1 60 120 2 72 200 3 65 130 4 72 205 5 67 180 6 69 180 7 68 193 8 69 195 9 61 115 10 62 140 11 5.5 160

You should be able copy and paste the data by highlighting the entire table.

Answer the following questions regarding the relationship.

(a) Using all 11 data pairs for height and weight, calculate the correlation coefficient. Round your answer to 3 decimal places.
r =

(b) Is there a significant linear correlation between these 11 data pairs?

YesNo    

(c) Using only the first 10 data pairs for height and weight, calculate the correlation coefficient. Round your answer to 3 decimal places.
r =

(d) Is there a significant linear correlation between these 10 data pairs?

YesNo    

(e) Which statement explains this situation?

The height for the last data pair must be an error.

The erroneous value from the last data pair ruined a perfectly good correlation.    

Despite the low correlation coefficient from part (a), there is probably a significant correlation between height and weight.

All of these are valid statements.

Answer 1

Given is a Height vs Weight - Erroneous Data

Also given is a scatter plot, regression line, and corresponding data for the height and weight of 11 randomly selected adults.

Index	Height (x)	Weight (y)
	Inches	Pounds
1	60	120
2	72	200
3	65	130
4	72	205
5	67	180
6	69	180
7	68	193
8	69	195
9	61	115
10	62	140
11	5.5	160

The scatter plot of the above data.

Scatterplot of weight (y) vs height (x) weight (y) 0 10 20 30 50 60 70 80 40 height (x)

Clearly we can see that all other 10 points falls in a line except for the 11th data in the data set so clearly the 11th data is erroneous.

Before we go on to solve the problems let us know a bit about correlation coefficient.

Correlation Coefficient

The correlation coefficient is a measure of degree of linear relationship between two variables x and y. It is denoted by r and is calculated by,

$r=\frac{Covariance(x,y)}{\sqrt{Variance(x)}\sqrt{Variance(y)}}=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}}$

-1<r<1

r=-1, High negative correlation between x and y

r=0, x and y are not correlated

r=1, High positive correlation between x and y

Coming back to our problem

(a) Here we need to calculate the correlation coefficient using all 11 data pairs for height and weight.

The table of calculations is provided below,

Index	Height (x)	Weight (y)	(xi-x̄)^2	(yi-ȳ)^2	(xi-x̄)*(yi-ȳ)
	Inches	Pounds
1	60	120	0.9111	2049.6174	43.2128
2	72	200	122.0031	1205.9854	383.5804
3	65	130	16.3661	1244.1634	-142.6957
4	72	205	122.0031	1578.2584	438.8079
5	67	180	36.5481	216.8934	89.0339
6	69	180	64.7301	216.8934	118.4885
7	68	193	49.6391	768.8032	195.3527
8	69	195	64.7301	883.7124	239.171
9	61	115	0.0021	2527.3444	-2.2874
10	62	140	1.0931	638.7094	-26.4226
11	5.5	160	3075.2016	27.8014	292.3949
Total	670.5	1818	3553.2276	11358.1822	1628.6364

$\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{670.5}{11}=60.9545$

$\overline{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}=\frac{1818}{11}=165.2727$

$r=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}}=\frac{1628.6364}{\sqrt{3553.2276}\sqrt{11358.1822}}$

$\Rightarrow r=0.256$

Hence the correlation coefficient using all 11 data pairs for height and weight is 0.256

(b) Now since r=0.256 which is close to 0 hence there is no significant linear correlation between these 11 data pairs.

So answer is No.

(c) Here we need to calculate the correlation coefficient using only the first 10 data pairs for height and weight.

The table of calculations is provided below,

Index	Height (x)	Weight (y)	(xi-x̄)^2	(yi-ȳ)^2	(xi-x̄)*(yi-ȳ)
	Inches	Pounds
1	60	120	42.25	2097.64	297.7
2	72	200	30.25	1169.64	188.1
3	65	130	2.25	1281.64	53.7
4	72	205	30.25	1536.64	215.6
5	67	180	0.25	201.64	7.1
6	69	180	6.25	201.64	35.5
7	68	193	2.25	739.84	40.8
8	69	195	6.25	852.64	73
9	61	115	30.25	2580.64	279.4
10	62	140	20.25	665.64	116.1
Total	665	1658	170.5	11327.6	1307

$\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{665}{10}=66.5$

$\overline{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}=\frac{1658}{10}=165.8$

$r=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}}=\frac{1307}{\sqrt{170.5}\sqrt{11327.6}}$

$\Rightarrow r=0.940$

Hence the correlation coefficient using only the first 10 data pairs for height and weight is 0.940

(d) Now since r=0.940 which is close to 1 hence there is a significant linear correlation between these first 10 data pairs.

So answer is Yes.

(e)

• Clearly the height for the last data pair must be an error because looking at the other heights, height cannot be 5.5
• Clearly the erroneous value from the last data pair ruined a perfectly good correlation because when we omitted the last pair the other 10 pairs had a correlation of 0.940.
• Despite the low correlation coefficient from part (a) (which was due to the erroneous pair), there is probably a significant correlation between height and weight (which is clear from the part c and d).

Hence all of these statements are valid statements.