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Diamond weight and price The weight (in carats) and the price (in millions of dollars) of the 9 most expensive diamonds in the world was collected from www.elitetraveler.com. Let the explanatory variable \(x=\) weight and the response variable \(y=\) price. The regression equation is \(\hat{y}=109.618+0.043 x\). a. Princie is a diamond whose weight is 34.65 carats. Use the regression equation to predict its price. b. The selling price of Princie is \(\$ 39.3\) million. Calculate the residual associated with the diamond and comment on its value in the context of the problem. c. The correlation coefficient is \(0.053 .\) Does it mean that a diamond's weight is a reliable predictor of its price?

Step by step solution

01

Identify the Variables

For part (a), we know that the weight of the diamond, which is our explanatory variable, is 34.65 carats. We have a regression equation: \( \hat{y} = 109.618 + 0.043x \). Here, \(x\) stands for the diamond's weight, and \(\hat{y}\) is the predicted price in millions.

02

Substitute and Solve (a)

Substitute \( x = 34.65 \) into the regression equation to predict the price:\[ \hat{y} = 109.618 + 0.043 \times 34.65 \]Calculate:\[ \hat{y} = 109.618 + 1.48995 = 111.10795 \]The predicted price of the Princie diamond is approximately $111.11 million.

03

Define Residual (b)

Residual is the difference between the observed price and the predicted price. For Princie, it is given that the observed price \( y = 39.3 \) million.

04

Calculate the Residual (b)

The residual for Princie is calculated as follows:\[ \text{Residual} = y - \hat{y} = 39.3 - 111.10795 \approx -71.80795 \]The residual is approximately -71.81 million, meaning the predicted price is much higher than the actual price.

05

Analyze Correlation (c)

The correlation coefficient \( r = 0.053 \) indicates a very weak correlation between the weight and price. This low value suggests that weight is not a reliable predictor of price.

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Key Concepts

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

Regression Analysis

Regression analysis is a statistical tool used to examine the relationship between two or more variables. In simple linear regression, this involves an explanatory variable (independent) and a response variable (dependent).

For example, in our exercise, the explanatory variable is the diamond's weight (\(x\)), and the response variable is its price (\(y\)). The goal is to determine how the explanatory variable influences the response variable. The relationship is represented by a regression equation, such as \(\hat{y} = 109.618 + 0.043x\). This equation suggests that for every additional carat in weight, the price increases by 0.043 million dollars.

• Regression equations help predict value changes based on variable changes.
• They offer insights into trends and data patterns, aiding in decision-making.

Correlation Coefficient

The correlation coefficient, denoted by \(r\), measures the strength and direction of a linear relationship between two variables on a scatter plot. This value ranges between -1 and 1.

In our diamond example, the correlation coefficient was \(r = 0.053\), indicating a very weak positive linear relationship between weight and price. A correlation so close to zero suggests that there’s almost no linear relationship.

• If \(r = 1\) or \(-1\), there's a perfect linear relationship.
• \(r = 0\) suggests no linear relationship at all.
• A weak correlation might imply other factors influence the response variable.

Thus, using weight alone to predict price isn't reliable, as evidenced by the low correlation.

Residuals

Residuals are the differences between observed values and the values predicted by a model. These provide insight into the model's accuracy in predicting outcomes.

Mathematically, it's calculated as: \[\text{Residual} = y - \hat{y}\], where \(y\) is the observed value, and \(\hat{y}\) is the predicted value. For the Princie diamond, the residual was calculated as \(-71.81\) million dollars, which indicates the prediction was much higher than the actual price.

• Large residuals highlight discrepancies or variability not captured by the model.
• Consistently large residuals could imply a model isn't the best fit.

Analyzing residuals helps in refining models and recognizing trends or outliers.

Predictive Modeling

Predictive modeling uses statistical techniques to define relationships and predict future outcomes based on datasets. Regression analysis often forms the backbone of these models.

In our scenario, predictive modeling aims to forecast diamond prices based on weight. With the regression equation, we predict how changes in weight affect prices. However, the weak correlation coefficient suggests weight alone is inadequate for accurate predictions.

• Reliable predictive models account for multiple influencing factors, enhancing accuracy.
• Models improve over time with more data and better statistical methods.
• They are crucial in various industries for planning and strategy.

Consistently improving models ensures they stay relevant and useful for predictions.