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Time at the table Does how long young children remain at the lunch table help predict how much they eat? Here are data on a random sample of 20 toddlers observed over several months. \({ }^{10}\) "Time" is the average number of minutes a child spent at the table when lunch was served. "Calories" is the average number of calories the child consumed during lunch, calculated from careful observation of what the child ate each day. Some computer output from a least-squares regression analysis on these data is shown below. $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 560.65 & 29.37 & 19.09 & 0.000 \\ \text { Time } & -3.0771 & 0.8498 & -3.62 & 0.002 \\ S=23.3980 & R-S q=42.1 \% & R-S q(a d j)=38.9 \% \end{array} $$ (a) What is the equation of the least-squares regression line for predicting calories consumed from time at the table? Interpret the slope of the regression line in context. Does it make sense to interpret the \(y\) intercept in this case? Why or why not? (b) Explain what the value of \(s\) means in this setting. (c) Do these data provide convincing evidence at the \(\alpha=0.01\) level of a linear relationship between time at the table and calories consumed in the population of toddlers? Assume that the conditions for performing inference are met.

Step by step solution

01

Identify the Regression Equation

The general form of a simple linear regression equation is \( \hat{y} = a + bx \), where \( \hat{y} \) is the predicted value, \( a \) is the y-intercept, \( b \) is the slope, and \( x \) is the predictor variable. From the given computer output, the regression equation can be written as \( \hat{y} = 560.65 - 3.0771x \), where \( y \) is calories and \( x \) is time.

02

Interpret the Slope

The slope of the regression line is \(-3.0771\). This means that for each additional minute a child spends at the lunch table, the average number of calories consumed decreases by approximately 3.0771 calories. In context, this suggests a negative relationship: as the time spent at the table increases, calorie consumption decreases.

03

Interpret the Y-Intercept

The y-intercept is 560.65, which represents the predicted calorie consumption when the time at the table is zero. In context, it does not make sense to interpret this value practically because a child cannot consume calories without spending any time at the table.

04

Understand the Standard Deviation of Residuals

The value of \( s = 23.3980 \) represents the standard deviation of the residuals, or the average distance that the observed values fall from the regression line. In this context, it means that, on average, the actual number of calories consumed deviate by about 23.3980 calories from the regression line prediction.

05

Test for Linear Relationship Using t-Test

To determine if there's convincing evidence of a linear relationship, we refer to the t-statistic and p-value for the slope of time. The hypothesis test is structured as follows:- Null hypothesis (\( H_0 \)): There is no linear relationship between time and calories (\( \beta = 0 \)).- Alternative hypothesis (\( H_a \)): There is a linear relationship (\( \beta eq 0 \)).Given t = -3.62 and p = 0.002 (from the output), the p-value of 0.002 is less than the significance level \( \alpha = 0.01 \). Hence, we reject the null hypothesis and conclude there is sufficient evidence to support a linear relationship at the 0.01 significance level.

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Key Concepts

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

Slope Interpretation

The slope of a regression line in a linear model quantifies the relationship between the predictor variable and the response variable. In this exercise, we're looking at how the time toddlers spend at the lunch table affects how many calories they consume. The given slope is \(-3.0771\). This means for each additional minute a child spends at the table, the average number of calories consumed decreases by approximately 3.0771 calories. Understanding the slope:

• A negative slope, like we have here, suggests an inverse relationship: as one variable increases, the other decreases.
• In this context, as children spend more time at the table, they tend to eat fewer calories on average.

This could be due to factors such as becoming less interested in eating as time goes on or potentially engaging with other activities like playing. The slope gives us insight into the nature of the relationship between time and caloric intake.

Y-Intercept

The y-intercept in a regression equation is the predicted value of the response variable when the predictor variable equals zero. In our regression equation \( \hat{y} = 560.65 - 3.0771x \), the y-intercept is 560.65. This value represents the predicted average calorie intake when the time at the table is zero.However, interpreting this in practical terms can be tricky and often doesn't make sense. Here’s why:

• Realistically, eating food while spending no time at the table is not possible.
• The y-intercept exists as part of the equation mathematically, but its practical usefulness depends on the situation.

This means in this context, the y-intercept doesn't lend itself to meaningful interpretation, as toddlers cannot consume calories without spending some time at the table.

Standard Deviation of Residuals

The standard deviation of residuals, denoted as \(s\), measures how much the observed values deviate from the predicted values on average. For this exercise, \(s = 23.3980\). Here, the residuals refer to the differences between the actual calories consumed by the toddlers and the calories predicted by our regression model.How to interpret this value:

• The average prediction error made by the regression line is about 23.3980 calories.
• This means that each child's observed caloric intake typically varies from the prediction by approximately 23.4 calories.

Understanding the standard deviation of residuals helps to gauge the accuracy of the predictions made by the regression line. Smaller values are favorable and indicate that the model predicts the response variable more accurately.

Hypothesis Testing

Hypothesis testing in the context of linear regression helps determine if there is statistically significant evidence of a relationship between the predictor and response variables. In our scenario, we want to understand if the time toddlers spend at the table is related to their caloric intake.We structure our hypotheses as follows:

• Null hypothesis \((H_0)\): There's no linear relationship between time and calories, expressed as \(\beta = 0\).
• Alternative hypothesis \((H_a)\): There is a linear relationship, expressed as \(\beta eq 0\).

The t-statistic from our data is -3.62 with a p-value of 0.002. Comparing this p-value to our significance level \(\alpha = 0.01\), we see that 0.002 is smaller.This result leads us to reject the null hypothesis, suggesting there's strong evidence of a linear relationship between time spent at the table and caloric intake among toddlers. Consequently, with this information, one might consider further exploration or potential dietary guidelines based on meal duration.