Research performed at NASA and led by Emily R. Morey-Holton...

Research performed at NASA and led by Emily R. Morey-Holton measured the lengths of the right humerus and right tibia in 11 rats that were sent to space on Spacelab Life Sciences $2 .$ The following data were collected. $$\begin{array}{|ll|ll} \text { Right } & \text { Right } & \text { Right } & \text { Right } \\ \text { Humerus } & \text { Tibia } & \text { Humerus } & \text { Tibia } \\ (\mathrm{mm}) & (\mathrm{mm}) & (\mathrm{mm}) & (\mathrm{mm}) \\ \hline 24.80 & 36.05 & 25.90 & 37.38 \\ \hline 24.59 & 35.57 & 26.11 & 37.96 \\ \hline 24.59 & 35.57 & 26.63 & 37.46 \\ \hline 24.29 & 34.58 & 26.31 & 37.75 \\ \hline 23.81 & 34.20 & 26.84 & 38.50 \\ \hline 24.87 & 34.73 & & \\ \hline \text {Source} \text { NASA Life Sciences Data Archive } & & \end{array}$$ (a) Draw a scatter diagram treating the length of the right humerus as the explanatory variable and the length of the right tibia as the response variable. (b) Compute the linear correlation coefficient between the length of the right humerus and the length of the right tibia. (c) Does a linear relation exist between the length of the right humerus and the length of the right tibia? (d) Convert the data to inches $(1 \mathrm{mm}=0.03937 \text { inch })$, and recompute the linear correlation coefficient. What effect did the conversion from millimeters to inches have on the linear correlation coefficient?

Research performed at NASA and led by Emily R. Morey-Holton measured the lengths of the right humerus and right tibia in 11 rats that were sent to space on Spacelab Life Sciences $2 .$ The following data were collected.
$$\begin{array}{|ll|ll}
\text { Right } & \text { Right } & \text { Right } & \text { Right } \\
\text { Humerus } & \text { Tibia } & \text { Humerus } & \text { Tibia } \\
(\mathrm{mm}) & (\mathrm{mm}) & (\mathrm{mm}) & (\mathrm{mm}) \\
\hline 24.80 & 36.05 & 25.90 & 37.38 \\
\hline 24.59 & 35.57 & 26.11 & 37.96 \\
\hline 24.59 & 35.57 & 26.63 & 37.46 \\
\hline 24.29 & 34.58 & 26.31 & 37.75 \\
\hline 23.81 & 34.20 & 26.84 & 38.50 \\
\hline 24.87 & 34.73 & & \\
\hline \text {Source} \text { NASA Life Sciences Data Archive } & &
\end{array}$$
(a) Draw a scatter diagram treating the length of the right humerus as the explanatory variable and the length of the right tibia as the response variable.
(b) Compute the linear correlation coefficient between the length of the right humerus and the length of the right tibia.
(c) Does a linear relation exist between the length of the right humerus and the length of the right tibia?
(d) Convert the data to inches $(1 \mathrm{mm}=0.03937 \text { inch })$, and recompute the linear correlation coefficient. What effect did the conversion from millimeters to inches have on the linear correlation coefficient?

Key Concepts

Effect of Unit Conversion on Correlation

Unit conversion, such as changing measurements from millimeters to inches, involves a linear scaling of the data. This transformation does not alter the underlying linear relationships, and therefore, the linear correlation coefficient remains invariant. This concept highlights that correlation is independent of the units of measurement used.

Linear Relationship

A linear relationship between two variables exists when a change in one variable is associated with a proportional change in the other variable. This concept is central to inferential statistics and regression analysis, allowing for predictions and interpretations based on the observed data.

Explanatory and Response Variables

Explanatory (independent) variables and response (dependent) variables are used to denote which variable is presumed to influence or predict the outcome. In analysis, the explanatory variable is plotted on the x-axis and is considered the cause, while the response variable on the y-axis is the effect or outcome of interest.

Scatter Diagram

A scatter diagram is a graphical representation that displays the relationship between two quantitative variables. It plots one variable on the horizontal axis (commonly the explanatory variable) and the other on the vertical axis (commonly the response variable), which helps visualize patterns, trends, or potential correlations between them.

Linear Correlation Coefficient

The linear correlation coefficient is a statistic that quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative linear relationship, and values near 0 suggest little to no linear correlation.

Transcript

00:01 Hello viewers.

00:02 In this problem we have a data representation of the measured length of right humerus along with the right tavia in 11 rats that were sent to space on space life sciences too.

00:14 And this is the data that was collected.

00:18 We have to draw a scattered diagram creating the length of the right humerus as an explanatory variable and the length of right tibia as a response variable.

00:29 The scatter plot is a diagram that is used to view.

00:32 Visualize the direction and strength of the linear relationship between any two variables.

00:38 The explanatory variable that is denoted by x is plotted along the horizontal axis.

00:45 So x is the explanatory variable which is right humorous.

00:56 While the response variable that is denoted by y is plotted along the vertical axis.

01:02 So y will be the response variable which is right t.

01:09 The total number of observation is 11.

01:18 Therefore, if each observation is plotted as a point on the graph to obtain the scatterplot, the scattered plot will be obtained like this.

01:28 Here, x represents the right humorous and y represents the right tibium.

01:35 Now, in part b we have to compute the linear correlation coefficient between the length of the right humorous and the length of the right tibia.

01:44 So, the correlation coefficient r measures the correlation coefficient r measures the degree of strength of linear relationship between any two variables and it is given by the formula sum of x i minus x bar divided by x x multiplied by y i minus y bar divided by s y and this is n minus one your n represents the number of observations on substituting the values that is x i will be the length of the right humorous and y i is will be the length of the the right tibium.

02:22 We will get the correlation coefficient as 0 .951.

02:29 So this is the part b of a problem...

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