TI-84 output from a linear model for predicting arm span (in...

TI-84 output from a linear model for predicting arm span (in centimeters) from height (in inches) is given in the figure. Summary statistics are also provided. $$ \begin{array}{|lrc|} \hline & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Height, } x & 63.59 & 3.41 \\ \hline \text { Arm span, } y & 159.86 & 8.10 \\ \hline \end{array} $$ To do parts a through $\mathrm{c}$, assume that the association between arm span and height is linear. a. Report the regression equation, using the words height and arm span, not $x$ and $y$, employing the output given. b. Verify the slope by using the formula $b=r \frac{s_{y}}{s_{x}}$. c. Verify the $y$ -intercept using $a=\bar{y}-b \bar{x}$. d. Using the regression equation, predict the arm span (in centimeters) for someone 64 inches tall.

TI-84 output from a linear model for predicting arm span (in centimeters) from height (in inches) is given in the figure. Summary statistics are also provided.
$$
\begin{array}{|lrc|}
\hline & \text { Mean } & \text { Standard Deviation } \\
\hline \text { Height, } x & 63.59 & 3.41 \\
\hline \text { Arm span, } y & 159.86 & 8.10 \\
\hline
\end{array}
$$
To do parts a through $\mathrm{c}$, assume that the association between arm span and height is linear.
a. Report the regression equation, using the words height and arm span, not $x$ and $y$, employing the output given.
b. Verify the slope by using the formula $b=r \frac{s_{y}}{s_{x}}$.
c. Verify the $y$ -intercept using $a=\bar{y}-b \bar{x}$.
d. Using the regression equation, predict the arm span (in centimeters) for someone 64 inches tall.

Key Concepts

Simple Linear Regression

Simple linear regression is a method used to establish and quantify the linear relationship between two variables, typically one independent and one dependent variable. It involves estimating a line (the regression line) that best fits the data by minimizing the discrepancies between observed values and predicted values, which is fundamental in predictive analysis and inferential statistics.

Regression Equation

The regression equation is the mathematical representation of the relationship between the independent and dependent variables, usually written in the form 'dependent variable = intercept + slope × independent variable'. This equation can then be used to make predictions or to understand how changes in the independent variable affect the dependent variable.

Slope Calculation (b = r × (s?_y / s?_x))

The slope in a regression context indicates the amount by which the dependent variable is expected to change for a one-unit increase in the independent variable. It can be calculated using the formula b = r × (s_y / s_x), where r is the correlation coefficient between the variables and s_y and s_x are their respective standard deviations. This ties the strength and direction of the relationship to the variability observed in the data.

Intercept Calculation (a = ? - b × x?)

The intercept of a regression line is the predicted value of the dependent variable when the independent variable is zero. It is calculated using the equation a = ? - b × x?, where ? and x? represent the means of the dependent and independent variables, respectively. The intercept provides a baseline value from which the effect of the independent variable begins.

Prediction Using the Regression Model

Prediction in linear regression involves substituting a specific value of the independent variable into the regression equation to estimate the corresponding dependent variable. This allows for practical applications such as forecasting or making decisions based on predicted outcomes, ensuring that the model's insights can be applied to real-world scenarios.

Transcript

0:00 Hello.

00:01 So in part 8, we have the one, in a given problem, the arm span is going to be predicted using the height.

00:08 So therefore, the arm span is the dependent variable.

00:11 Height is the independent variable.

00:13 So in the given summary statistics, we have the slope of the regression equation is going to be b, which is approximately 2 .25.

00:21 The y intercept is a, which is approximately 16 .80.

00:25 So our regression equation then, in terms of height and arm span is going to be the.

00:30 Predicted arm span is going to be equal to 16 .80 plus 2 .25 times the height.

00:37 And then for part b, we are looking to verify the slope using the formula that b is equal to r times sy over x.

00:51 So this is the standard deviation of the height of x is 3 .41 and the standard deviation of the arm span of y is 8 .10.

01:02 So then we get that b is going to be equal to 0 .95 times 8 .10 divided by 3 .41, which is going to be approximately equal to 2 .25.

01:23 So the value of the slope in the given statistics here is about 2 .25, which is equal to the value of the slope obtained by using the formula.

01:32 B is equal to r times standard deviations.

01:35 Of y divided by the standard deviation of x.

01:38 Therefore, it's verified that both slopes are equal...

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