91Ó°ÊÓ

The article "Promoting Healthy Choices: Information versus Convenience" (Amer. Econ. J.: Applied Econ., 2010: 164 - 178) reported on a field experiment at a fast-food sandwich chain to see whether calorie information provided to patrons would affect calorie intake. One aspect of the study involved fitting a multiple regression model with 7 predictors to data consisting of 342 observations. Predictors in the model included age and dummy variables for gender, whether or not a daily calorie recommendation was provided, and whether or not calorie information about choices was provided. The reported value of the \(F\) ratio for testing model utility was \(3.64\). a. At significance level .01, does the model appear to specify a useful linear relationship between calorie intake and at least one of the predictors? b. What can be said about the \(P\)-value for the model utility \(F\) test? c. What proportion of the observed variation in calorie intake can be attributed to the model relationship? Does this seem very impressive? Why is the \(P\)-value as small as it is? d. The estimated coefficient for the indicator variable calorie information provided was \(-71.73\), with an estimated standard error of \(25.29\). Interpret the coefficient. After adjusting for the effects of other predictors, does it appear that true average calorie intake depends on whether or not calorie information is provided? Carry out a test of appropriate hypotheses.

Step by step solution

01

Identify Null and Alternative Hypotheses for the F-test

The null hypothesis for the F-test is that the model with all predictors does not explain the variability in the dependent variable better than a model with no predictors. The alternative hypothesis is that at least one predictor variable explains some of the variability.\(H_0: \beta_1 = \beta_2 = \ldots = \beta_7 = 0 \H_a: \text{at least one } \beta_i eq 0\)

02

Determine Critical Value at Significance Level 0.01

The F-statistic follows an F-distribution with degrees of freedom corresponding to the number of predictors ( k = 7) and the number of observations minus the number of predictors minus one ( 342 - 7 - 1 = 334 ) . Find the critical value from F-distribution tables with significance level 0.01 and degrees of freedom 7 and 334.

03

Compare F-value to Critical F-value

With an F-ratio of 3.64, compare it to the critical value obtained in Step 2. If 3.64 exceeds this critical value, we reject the null hypothesis that the model is not useful.

04

Interpret the P-value

Given the F-ratio of 3.64 results in significance at the 0.01 level, the P-value for this test is less than 0.01. This implies strong evidence against the null hypothesis, supporting the claim that the model is useful.

05

Calculate the Proportion of Variability Explained

The proportion of variability explained by the model is given by the coefficient of determination, \( R^2 \). This can be estimated using the F-ratio:\[R^2 = \frac{F}{F + (N - k - 1)/k}\quad \rightarrow \quad R^2 = \frac{3.64}{3.64 + (342 - 7 - 1)/7}\]

06

Evaluate R-squared for Practical Significance

After calculating \( R^2 \), compare it to common benchmarks (e.g., 0.25, 0.50, 0.75) to assess practical significance. A small \( R^2 \) might still have statistical significance due to the small P-value, especially with a large sample size as here.

07

Interpret Coefficient for Calorie Information Provided

The coefficient \(-71.73\) suggests that providing calorie information decreases calorie intake by about 71.73 units, controlling for other predictors in the model. This is a significant reduction in calorie intake.

08

Hypothesis Test on Calorie Information Coefficient

Test the hypothesis that the coefficient for calorie information provided is zero, which means it does not affect the response:\(H_0: \beta = 0 \H_a: \beta eq 0\)Calculate the t-statistic using \(T = \frac{-71.73}{25.29} \approx -2.84\) and compare it to the critical t-value at 0.01 significance level with appropriate degrees of freedom.

09

Draw Conclusion from the T-test

If the absolute t-value exceeds the critical t-value, reject the null hypothesis. This indicates the true average calorie intake is related to whether calorie information is provided, adjusting for other factors.

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Key Concepts

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

Field Experiment

In research, a field experiment is a study conducted in a real-world setting rather than in a controlled laboratory environment. This approach allows researchers to observe natural behaviors and reactions, increasing the external validity of the findings. The study mentioned in the exercise was carried out in a fast-food sandwich chain, making it a field experiment because it took place in a natural setting where subjects freely made choices.

Field experiments often aim to understand the effect of a particular intervention in realistic situations. In this example, researchers wanted to know how providing calorie information influenced the calorie intake of customers at the sandwich chain. By observing customers in their usual eating environment, researchers could gather genuine data reflecting actual behavior.

Key aspects of conducting field experiments:

• Researchers do not control all external variables, unlike in a lab setting.
• Participants are often unaware they are part of an experiment to reduce bias and mimic real decision-making.
• The results can be more applicable to everyday settings, aiding policy decisions and real-world applications.

This approach can provide insights that would be difficult to achieve in artificial laboratory settings.

Dummy Variables

In multiple regression analysis, dummy variables are used to incorporate categorical data into the model. Dummy variables transform qualitative attributes into a format that can be analyzed quantitatively. This is essential when a predictor should represent a category like gender, presence or absence of a condition, or any other binary characteristic.

For instance, in the exercise discussed, gender, and whether calorie information or calorie recommendations were provided, are used as dummy variables. They help decipher the influence of these categorical factors on calorie intake.

Each category is typically represented by a 0 or 1:

• A category assigned '1' might indicate presence (e.g., calorie information provided).
• A '0' would indicate absence (e.g., calorie information not provided).

This binary transformation assists the regression model in comprehending how different categories impact the dependent variable. The resulting coefficients of dummy variables tell us the expected change in the dependent variable, calorie intake in this case, moving from one category to the other while all else is constant.

F-test

The F-test is a statistical procedure used to determine the overall significance of a regression model. It assesses whether at least one predictor variable has a significant linear relationship with the dependent variable. In the exercise context, an F-test evaluates the entire regression model with the given predictors to ascertain its utility.

In our example, the null hypothesis for the F-test is that none of the predictor variables explain the variability in calorie intake. The alternative hypothesis is that at least one does. The computed F value in the example is 3.64, and by comparing this value to a critical value from F-distribution tables, researchers can decide whether to reject the null hypothesis.

The process typically involves:

• Determining the F-ratio using the variance explained by the model and the variance unexplained (error variance).
• Comparing this F-statistic to a critical value based on the significance level and degrees of freedom.
• If the F-statistic exceeds the critical value, the model is considered useful, implying at least one predictor significantly relates to the dependent variable.
  
  Thus, the F-test provides a mechanism to determine the utility of a model before diving deeper into individual predictors.

Hypothesis Testing

Hypothesis testing in statistics is a method of making decisions or inferences about the characteristics of a population based on sample data. It begins with stating a null hypothesis ( H_0) and an alternative hypothesis ( H_a). The null hypothesis typically posits no effect or relationship, while the alternative suggests the contrary. In the exercise, hypothesis testing is used to assess whether the coefficient of providing calorie information in the regression model significantly influences calorie intake.

To carry out hypothesis testing:

• Select the appropriate test statistic based on the hypothesis and data structure, such as t-test for individual coefficients or F-test for the model.
• Calculate the test statistic value, in this case, the t-value was calculated using the estimate and its error.
• Determine the critical value from a statistical distribution (e.g., t-distribution).
• Compare the test statistic to the critical value or use the p-value, which indicates the probability of observing the data if H_0 is true.
• A small p-value, typically less than 0.05 or 0.01, leads to the rejection of the null hypothesis, indicating a statistically significant effect.
  
  This rigorous procedure enables researchers to draw conclusions about relationships or effects within data, such as figuring out the influence of calorie information on intake. By successfully refuting the null hypothesis, researchers provide evidence supporting the alternative hypothesis—that calorie information significantly reduces calorie intake according to the given data.