(Refer to Problem 28, Section 4.1 ) A study on antibiotic...

(Refer to Problem $28,$ Section 4.1 ) A study on antibiotic use among children in Manitoba, Canada, gave the following data for the number of prescriptions per 1000 children $x$ years after 1995. (TABLE CAN'T COPY) (a) Find the least-squares regression line, treating year as the explanatory variable and prescriptions as the response variable. (b) Interpret the slope and intercept, if appropriate. (c) Predict the number of prescriptions per 1000 children in Manitoba, Canada, in $2002(x=7)$. (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the number of prescriptions in Manitoba, Canada, in $2010 ?$ Why or why not?

(Refer to Problem $28,$ Section 4.1 ) A study on antibiotic use among children in Manitoba, Canada, gave the following data for the number of prescriptions per 1000 children $x$ years after 1995. (TABLE CAN'T COPY)
(a) Find the least-squares regression line, treating year as the explanatory variable and prescriptions as the response variable.
(b) Interpret the slope and intercept, if appropriate.
(c) Predict the number of prescriptions per 1000 children in Manitoba, Canada, in $2002(x=7)$.
(d) Draw the least-squares regression line on the scatter diagram of the data.
(e) Would it be reasonable to use the least-squares regression line to predict the number of prescriptions in Manitoba, Canada, in $2010 ?$ Why or why not?

Key Concepts

Extrapolation vs. Interpolation

Interpolation refers to using the regression model to predict values within the range of the observed data, where the estimates are generally reliable. Extrapolation involves predicting values outside the range of the observed data, which can be risky because the established relationship may not hold beyond the data’s limits, leading to potentially inaccurate predictions.

Prediction Using Regression

Once a regression model is established, it can be used to predict the response variable for given values of the explanatory variable. This involves substituting a specific value into the regression equation to estimate the corresponding outcome, making regression a powerful tool for forecasting and decision-making based on observed trends.

Interpretation of Slope and Intercept

The slope in a regression equation represents the rate of change in the response variable for each one-unit change in the explanatory variable, indicating the strength and direction of the relationship. The intercept represents the predicted value of the response variable when the explanatory variable is zero, though its interpretation may depend on whether a value of zero is meaningful in the context of the data.

Explanatory and Response Variables

In regression analysis, the explanatory (or independent) variable is the variable that is used to predict or explain changes in another variable, while the response (or dependent) variable is the outcome being predicted. Understanding which variable plays each role is crucial for correctly setting up the model and interpreting the relationship between them.

Least-Squares Regression

The least-squares regression is a statistical method used to find a straight line that best fits a set of data by minimizing the sum of the squared differences (residuals) between the observed values and those predicted by the line. This method is central to linear regression analysis and provides a way to model the relationship between an independent variable (predictor) and a dependent variable (response).

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