91Ó°ÊÓ

Measurements on young children in Mumbai, India, found this least-squares line for predicting height \(y\) from \(\operatorname{arm} \operatorname{span} x:\) $$\hat{y}=6.4+0.93 x$$ Measurements are in centimeters \((\mathrm{cm})\). In addition to the regression line, the report on the Mumbai measurements says that \(r^{2}=0.95\). This suggests that (a) although arm span and height are correlated, arm span does not predict height very accurately. (b) height increases by \(\sqrt{0.95}=0.97 \mathrm{~cm}\) for each additional centimeter of arm span. (c) \(95 \%\) of the relationship between height and arm span is accounted for by the regression line. (d) \(95 \%\) of the variation in height is accounted for by the regression line. (e) \(95 \%\) of the height measurements are accounted for by the regression line.

Step by step solution

01

Understand the Given Information

We are given a least-squares regression equation \( \hat{y} = 6.4 + 0.93x \), where \( \hat{y} \) is the predicted height and \( x \) is the arm span, both in centimeters. We are also given \( r^2 = 0.95 \). Our task is to understand what this \( r^2 \) value indicates about the relationship between arm span and height.

02

Interpret \( r^2 \) Value

The \( r^2 \) value, also known as the coefficient of determination, represents the proportion of the variance in the dependent variable (height) that is predictable from the independent variable (arm span) using the regression line. In this context, \( r^2 = 0.95 \) means 95% of the variation in height can be explained or accounted for by the variation in arm span.

03

Evaluate the Given Choices

Let's evaluate each option based on our understanding:- (a) This statement is incorrect because a high \( r^2 \) value (0.95) indicates that arm span predicts height very accurately.- (b) This statement is mathematically incorrect; the increase in height per centimeter of arm span is given by the slope (0.93 cm), not \( \sqrt{0.95} \).- (c) This option is misleading as it suggests a direct relationship between the observation values; instead, it should refer to the variance, not the relationship itself.- (d) This statement correctly reflects the interpretation of \( r^2 \); 95% of the variation in height is indeed accounted for by the regression line.- (e) This option suggests an understanding of individual measurements, which is incorrect when referring to the meaning of \( r^2 \).

04

Select the Correct Choice

From the evaluation, option (d) correctly states the implication of the \( r^2 \) value, that 95% of the variation in height is accounted for by the regression line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least-Squares Regression

Least-squares regression is a statistical method used to determine a line of best fit for a set of data. This line minimizes the sum of the squares of the vertical distances (residuals) between the observed values and the predicted values on the line. Think of it as finding the "best" line that goes as closely as possible through all your data points. Let's say we have data of arm span and height of children; our goal is to predict height based on arm span. The least-squares regression equation derived from this data could look like \( \hat{y} = 6.4 + 0.93x \). Here, \( \hat{y} \) is the predicted height, and \( x \) represents the arm span.

• "6.4" is the y-intercept, meaning if the arm span were zero, the predicted height would be 6.4 cm, which isn't meaningful in this context but is mathematically essential.
• "0.93" is the slope, which indicates that for every additional centimeter in arm span, the height increases by 0.93 cm.

This method helps us make predictions and understand the relationship between two variables.

Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), is a vital concept in regression analysis. It tells us how well the independent variable (like arm span in our case) explains the variation in the dependent variable (like height). If \( r^2 \) is close to 1, it means that a large portion (or percentage) of the variance in the predicted outcome (height) is explained by the model (arm span).For example, in the given case, \( r^2 = 0.95 \) implies that 95% of the variation in children's height can be explained by their arm span.

• This means that the regression model is very accurate in predicting height from arm span since only 5% of the variance is left unexplained.
• Thus, high \( r^2 \) values suggest a strong correlation between the variables and a good fit of the model.

Predicting Height

Predicting height using regression analysis is a practical application of mathematical modeling. By understanding the relationship between arm span and height, we can make educated predictions about one variable when we know the other.Using the regression equation \( \hat{y} = 6.4 + 0.93x \), if a child's arm span is 100 cm, you can predict their height by substituting \( x \) with 100: \(\hat{y} = 6.4 + 0.93 imes 100 = 99.4 \, \text{cm}\)This calculation means we estimate the child's height to be approximately 99.4 cm.

• This approach can be useful in many fields where straightforward data relationships help make reliable predictions.
• However, remember that predictions are based on past data, so there might be slight inaccuracies due to variations not captured in the model.

Variance Explanation

Variance explanation in the context of regression analysis refers to how much of the variability in a dependent variable is accounted for by the independent variable. In simpler terms, it helps us understand how much of what we see in our data can be explained by the factors we're analyzing.In our exercise, where \( r^2 = 0.95 \), this means 95% of the height variance among children can be explained by their arm span.

• This high value indicates that the regression model we have is very effective at explaining the variation in height based on arm span.
• The remaining 5% variance is due to other factors that the model does not account for, such as genetic factors, nutrition, or measurement errors.

Understanding variance explanation helps in assessing whether our model is suitable for prediction or if adjustments or additional factors should be considered.