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```
> fit <- lm(taste ~ Acetic + H2S + Lactic, data=cheddar)
> summary(fit)
Cal...
Sep 12, 2024
Solution by Steps
step 1
The linear regression model is given by the formula taste=β0+β1⋅Acetic+β2⋅H2S+β3⋅Lactic. Here, β0 is the intercept, and β1,β2,β3 are the coefficients for the independent variables
step 2
The coefficients from the output are: β0=−28.8768, β1=0.3277, β2=3.9118, and β3=19.6705. This indicates how each independent variable affects the dependent variable "taste"
step 3
The significance of each coefficient can be assessed using the p-values. For H2S, p=0.00425 (significant), and for Lactic, p=0.03108 (significant). Acetic is not significant with p=0.94198
step 4
The overall model fit can be evaluated using the Multiple R-squared value, which is 0.6518. This means approximately 65.18% of the variability in "taste" is explained by the model
step 5
The F-statistic is 16.22 with a p-value of 3.81e−06, indicating that the model is statistically significant overall
Answer
The linear regression analysis shows that H2S and Lactic are significant predictors of taste, while Acetic is not. The model explains about 65.18% of the variability in taste.
Key Concept
Linear regression analysis helps in understanding the relationship between dependent and independent variables, assessing significance through p-values and model fit through R-squared.
Explanation
The output indicates which variables significantly affect the dependent variable and how well the model explains the variability in the data.
which statements are true with the output above
Solution by Steps
step 1
The statement that "if we fit a model that only used Acetic acid to predict taste then Acetic would not be a statistically significant predictor of taste" is true, as indicated by the output where the p-value for Acetic is 0.94198, which is greater than 0.05
step 2
The statement that "the current model explains about 65% of the variation in taste" is true, as the Multiple R-squared value is 0.6518, which indicates that approximately 65.18% of the variability in taste is explained by the model
step 3
The statement "increasing the value of Lactic acid by one unit would increase the expected taste score by 19.67" is true, as the coefficient for Lactic is 19.6705, indicating the expected change in taste for a one-unit increase in Lactic acid
step 4
The statement "it is possible that in a population of cheeses where H2S and Lactic acid are held at fixed levels that changing the level of Acetic acid doesn't affect taste" is true, as the lack of significance of Acetic acid suggests that it may not have an effect when controlling for H2S and Lactic acid
Answer
All statements are true based on the output provided.
Key Concept
Understanding statistical significance and the interpretation of regression coefficients.
Explanation
The analysis of the regression output shows which predictors are significant and how much variation in the response variable is explained by the model.
Solution by Steps
step 1
The first statement suggests that the coefficients for H2S and Acetic acid can be individually plausible but not jointly reasonable. This is consistent with the confidence ellipse indicating that the combination of these coefficients falls outside the plausible region when considered together
step 2
The second statement contradicts the first, suggesting that both coefficients are not plausible individually but are reasonable when considered jointly. This is not supported by the confidence ellipse, which indicates that the joint consideration is crucial
step 3
The third statement claims that there are no combinations of parameters that are unreasonable according to both individual confidence intervals (CIs) but reasonable according to the confidence ellipse. This is incorrect as the ellipse provides a joint confidence region
step 4
The fourth statement correctly asserts that there are no combinations of parameters that are reasonable according to both individual CIs but unreasonable according to the confidence ellipse, which aligns with the interpretation of the graph
Answer
D
Key Concept
Understanding joint vs. individual parameter estimates in regression analysis
Explanation
The confidence ellipse represents the joint confidence region for parameter estimates, while individual confidence intervals assess parameters separately. This distinction is crucial for interpreting the plausibility of parameter combinations.
Solution by Steps
step 1
The statement claims that Forward Selection and Backward Selection will always arrive at the same best model if they consider the same set of variables. However, this is not necessarily true because these methods can lead to different models due to their different approaches to variable selection
step 2
Forward Selection adds variables based on their significance, while Backward Selection removes variables based on their insignificance. This difference in methodology can result in different final models, especially in the presence of multicollinearity or when variables have similar predictive power
step 3
Therefore, the statement is false as the two strategies do not guarantee the same best model even when starting with the same set of variables
Answer
B
Key Concept
Variable selection methods in regression analysis
Explanation
Forward and Backward Selection can yield different models due to their distinct approaches to adding or removing variables.
Multicollinearity is more likely to be a problem when analyzing data that have come from a designed experiment than when analyzing data from an observational study. Is the statement true?
Solution by Steps
step 1
The statement claims that multicollinearity is more likely to be a problem in designed experiments than in observational studies. In designed experiments, variables are often controlled and manipulated, which can lead to high correlations among predictors
step 2
In observational studies, variables are measured as they naturally occur, which may result in less control over the relationships between predictors, potentially reducing the likelihood of multicollinearity
step 3
Therefore, the statement is generally considered false, as multicollinearity can arise in both types of studies, but it is not inherently more likely in designed experiments compared to observational studies
Answer
False
Key Concept
Multicollinearity in statistical analysis
Explanation
The statement is false because multicollinearity can occur in both designed experiments and observational studies, but it is not necessarily more prevalent in one than the other.
Solution by Steps
step 1
The model Y∼X∗B corresponds to fitting parallel lines, one for each level of the categorical variable B. This indicates an interaction between X and B
step 2
The model Y∼X+B corresponds to fitting a single line to both categories combined, meaning that the effect of X is the same across the levels of B
step 3
The model Y∼X corresponds to fitting completely separate lines for each level of the categorical variable B, indicating that B does not influence the slope of X
Answer
A
Key Concept
Understanding the relationship between predictors and response variables in regression models
Explanation
Different models can represent various relationships between continuous and categorical variables, affecting how we interpret the data.
Considering a line that describes how flowering changes with intensity of light, the slope term for plants under both early and late Time conditions is... (use 2 decimal places).The intercept term for plants given the "early" Time treatment is ... (give 2 decimal places)
Solution by Steps
step 1
The slope term for plants under both early and late Time conditions is represented by the coefficient of the Intensity variable in the regression output. From the output, the coefficient for Intensity is −0.040471
step 2
Rounding this value to two decimal places gives us −0.04
step 3
The intercept term for plants given the "early" Time treatment is represented by the coefficient of the (Intercept) in the regression output. From the output, the intercept is 71.3058333
step 4
Rounding this value to two decimal places gives us 71.31
Answer
Slope term: -0.04; Intercept term: 71.31
Key Concept
The slope indicates the change in the response variable (flowering) for a one-unit change in the predictor (Intensity), while the intercept represents the expected value of the response when the predictor is zero.
Explanation
The slope of -0.04 suggests that as light intensity increases, flowering decreases slightly, and the intercept of 71.31 indicates the expected flowering level under early time conditions when intensity is zero.
Solution by Steps
step 1
When adding extra predictors to a model, the complexity of the model increases, which can lead to a higher R-squared value. This is because R-squared measures the proportion of variance explained by the model, and more predictors typically explain more variance
step 2
The F-test assesses whether at least one predictor variable has a non-zero coefficient. Adding predictors can increase the likelihood of finding significant predictors, but it does not guarantee that the F-test will be significant
step 3
The t-tests evaluate the significance of individual predictors. With more predictors, the t-tests may also become more likely to be significant, but this is contingent on the actual data and relationships
step 4
However, making a model more complicated does not make it easier to interpret; in fact, it often makes interpretation more challenging due to the increased number of variables
Answer
D
Key Concept
Adding predictors increases model complexity and can lead to higher R-squared values, but it complicates interpretation.
Explanation
While R-squared will likely increase, the interpretability of the model decreases, and significance tests may or may not yield significant results depending on the data.
In a model with a three way interaction between three predictor variables how many parameters need to be estimated?
Solution by Steps
step 1
In a model with three predictor variables, each variable can have its main effect, and we also need to account for interactions. For a three-way interaction, we need to consider all combinations of the three predictors
step 2
The number of parameters to estimate includes the intercept, the main effects of each predictor, the two-way interactions, and the three-way interaction. If we denote the predictors as X1, X2, and X3, the parameters can be calculated as follows:
step 3
The total number of parameters is given by:
Total Parameters=1+3+3+1=8
where:
- 1 is for the intercept,
- 3 for the main effects (X1, X2, X3),
- 3 for the two-way interactions (X1X2, X1X3, X2X3),
- 1 for the three-way interaction (X1X2X3)
Answer
8 parameters need to be estimated in a model with a three-way interaction among three predictor variables.
Key Concept
Understanding the number of parameters in regression models with interactions is crucial for model specification.
Explanation
In a three-way interaction model, you estimate parameters for the intercept, main effects, two-way interactions, and the three-way interaction itself, totaling 8 parameters.
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