Question

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the height (m) of the soccer ball. What do you conclude about the relationship between time and height? What horrible mistake would be easy to make if the analysis is conducted without a scatterplot?

Time (sec)	0.0	0.2	0.4	0.6	0.8	1.0	1.2	1.4	1.6	1.8
Height (m)	0.0	1.7	3.1	3.9	4.5	4.7	4.6	4.1	3.3	2.1

Short answer

The value of r is equal to 0.450.

Since the p-value of 0.192 is greater than 0.05, there is not a significant linear correlation between the time (sec) and height (m).

The scatter plot is represented as,

Step-by-step solution

Given information

The table represents the time (sec) that has lapsed from the throw and the height (m) of the soccer ball.

Calculate the correlation coefficient

Let x represents the Time (sec).

Let y represent the Height (m).

The formula for computing the correlation coefficient (r) between the values of Time (sec) and Height (m) is as follows:

\(r = \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\)

The following calculations are done to compute the value of r:

x	y	xy	\({x^2}\)	\({y^2}\)
0	0	0	0	0
0.2	1.7	0.34	0.04	2.89
0.4	3.1	1.24	0.16	9.61
0.6	3.9	2.34	0.36	15.21
0.8	4.5	3.6	0.64	20.25
1	4.7	4.7	1	22.09
1.2	4.6	5.52	1.44	21.16
1.4	4.1	5.74	1.96	16.81
1.6	3.3	5.28	2.56	10.89
1.8	2.1	3.78	3.24	4.41
\(\sum x \)=9	\(\sum y \)=32	\(\sum {xy} \)=32.54	\(\sum {{x^2}} \)=11.4	\(\sum {{y^2}} \)=123.32

Substituting the above values, the value of r is obtained as,

\(\begin{aligned} r &= \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\\ &= \frac{{10\left( {32.54} \right) - \left( 9 \right)\left( {32} \right)}}{{\sqrt {10\left( {11.4} \right) - {{\left( 9 \right)}^2}} \sqrt {10\left( {123.32} \right) - {{\left( {32} \right)}^2}} }}\\ &= 0.450\end{aligned}\)

Therefore, the value of r is equal to 0.450.

Significance of r

Here, n=10.

If the value of the correlation coefficient lies between the critical values, then the correlation between the two variables is considered significant else, it is considered insignificant.

The critical values of r for n=10 and \(\alpha = 0.05\) are -0.632 and 0.632.

The corresponding p-value of r is equal to 0.192.

Since the computed value of r equal to 0.450 is greater than the larger critical value of 0.192, it can be said that the correlation between the two variables is insignificant.

Moreover, the p-value is greater than 0.05. This also implies that correlation is insignificant.

Therefore, there is not sufficient evidence to claim that there is a linear correlation between the time (sec) and the height (m).