Statistics on Marginal Distributions
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Questions and Answers

What is the equation of the least-squares regression line?

y = b1x + b0

What does the notation 'y' represent in the least-squares regression line?

A predicted value of y for a given value of x.

How do you determine the slope (b1) of the least-squares regression line?

b1 = r * sy / sx

What is the y-intercept (b0) of the least-squares regression line?

<p>b0 = y - b1 * x</p> Signup and view all the answers

A scatter diagram shows a positive linear relationship if the y-value increases whenever the x-value increases.

<p>True</p> Signup and view all the answers

A residual is the difference between observed and predicted values in regression analysis.

<p>True</p> Signup and view all the answers

What is the predicted head circumference of a child who is 24 inches tall using the regression line y = 0.081x + 15.1?

<p>17.04 inches</p> Signup and view all the answers

What does the slope of 0.081 represent in the context of the regression line relating height to head circumference?

<p>For every inch increase in height, the head circumference increases by 0.081 inches, on average.</p> Signup and view all the answers

What would be the least-squares regression line for the data set with x=3.5000, sx=2.5100, y=4.0500, sy=1.7085, and r=−0.9538?

<p>y = -0.649x + 6.322</p> Signup and view all the answers

Why is it not appropriate to interpret the y-intercept in the head circumference and height regression line?

<p>The y-intercept doesn't make sense since a child's height cannot be 0 inches.</p> Signup and view all the answers

What is the formula to calculate the residual?

<p>Residual = Observed - Predicted</p> Signup and view all the answers

The plot of the residuals shows no discernible pattern. What does this imply?

<p>A linear model is appropriate.</p> Signup and view all the answers

What does the coefficient of determination of 88.4% mean?

<p>The least squares regression line explains 88.4% of the variation in length of eruption.</p> Signup and view all the answers

What is the correlation between distance and sidereal year?

<p>0.987</p> Signup and view all the answers

Does a correlation coefficient of 0.987 imply a linear relation between distance and sidereal year?

<p>True</p> Signup and view all the answers

What is the equation of the least-squares regression line?

<p>y = 0.065x - 12.3</p> Signup and view all the answers

What does the residual plot indicate about the least-squares regression line?

<p>It is a good model because the residuals do not form a pattern.</p> Signup and view all the answers

Does the store brand appear to have more chips per cookie compared to the name brand?

<p>False</p> Signup and view all the answers

Which brand has a more consistent number of chips per cookie?

<p>The name brand has a more consistent number of chips.</p> Signup and view all the answers

What is meant by a marginal distribution?

<p>A marginal distribution is the distribution of the values of one variable without regard to the values of other variables.</p> Signup and view all the answers

What is a marginal distribution?

<p>A marginal distribution is a frequency or relative frequency distribution of either the row or column variable in a contingency table.</p> Signup and view all the answers

What is meant by a conditional distribution?

<p>A conditional distribution is the distribution of one variable given a fixed value of another variable.</p> Signup and view all the answers

What is meant by a conditional distribution?

<p>A conditional distribution lists the relative frequency of each category of the response variable, given a specific value of the explanatory variable in a contingency table.</p> Signup and view all the answers

Is ethnicity associated with opinion regarding immigration?

<p>Yes, ethnicity is associated with opinion regarding immigration. Hispanics are more likely to feel that immigration is a good thing for the country and much less likely to feel it is a bad thing.</p> Signup and view all the answers

What is the relative frequency marginal distribution for the row variable opinion?

<p>It is found by dividing the row total for each opinion by the table total.</p> Signup and view all the answers

What is the difference between univariate data and bivariate data?

<p>In univariate data, a single variable is measured on each individual. In bivariate data, two variables are measured on each individual.</p> Signup and view all the answers

What does it mean to say that two variables are positively associated?

<p>There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable increases.</p> Signup and view all the answers

What does it mean to say that two variables are negatively associated?

<p>There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable decreases.</p> Signup and view all the answers

True or false: Correlation implies causation.

<p>The statement is false. Correlation can only be used to imply causation as a result of a properly designed experiment.</p> Signup and view all the answers

Do the two variables have a linear relationship based on the scatter diagram described?

<p>The data points do not have a linear relationship because they do not lie mainly in a straight line.</p> Signup and view all the answers

If the relationship is linear, do the variables have a positive or negative association?

<p>The relationship is not linear.</p> Signup and view all the answers

Match the linear correlation coefficient to the scatter diagram.

<p>r=1</p> Signup and view all the answers

What is a residual?

<p>A residual is the difference between an observed value of the response variable y and the predicted value of y. If it is positive, then the observed value is greater than the predicted value.</p> Signup and view all the answers

Explain what each point on the least-squares regression line represents.

<p>Each point on the least-squares regression line represents the predicted y-value at the corresponding value of x.</p> Signup and view all the answers

If the linear correlation between two variables is negative, what can be said about the slope of the regression line?

<p>Negative</p> Signup and view all the answers

Will increasing the percentage of the population that has a cell phone decrease the violent crime rate?

<p>No</p> Signup and view all the answers

What might be a lurking variable between the percentage of the population with a cell phone and the violent crime rate?

<p>The economy</p> Signup and view all the answers

Would it be reasonable to use the least-squares regression line to predict the head circumference of a child who was 32 inches tall? Why?

<p>No—this height is outside the scope of the model.</p> Signup and view all the answers

The _______ , R2, measures the proportion of total variation in the response variable that is explained by the least squares regression line.

<p>coefficient of determination</p> Signup and view all the answers

Total deviation = _______ deviation + _______ deviation. Choose the correct answer below.

<p>Total deviation = unexplained deviation + explained deviation</p> Signup and view all the answers

A _______ is a scatter diagram with the residuals on the vertical axis and the explanatory variable on the horizontal axis.

<p>residual plot</p> Signup and view all the answers

Analyze the residual plot and identify which, if any, of the conditions for an adequate linear model is not met.

<p>The residual plot shows a U-shaped pattern.</p> Signup and view all the answers

Which of the conditions below might indicate that a linear model would not be appropriate?

<p>Patterned residuals</p> Signup and view all the answers

Match the coefficient of determination to the scatter diagram.

<p>R2=0.58 = Scatter diagram II R2=0.94 = Scatter diagram I R2=0.01 = Scatter diagram III</p> Signup and view all the answers

Is the point in blue (large point) influential?

<p>No, because the point does not significantly affect the least-squares regression line.</p> Signup and view all the answers

Does the point in blue seem to be influential?

<p>No, because the observation does not significantly affect the least-squares regression line's slope and/or y-intercept.</p> Signup and view all the answers

What type of relation appears to exist between time between eruptions and length of eruption?

<p>Linear, positive association</p> Signup and view all the answers

Does the residual plot confirm that the relation between time between eruptions and length of eruption is linear?

<p>Yes. The plot of the residuals shows a discernible pattern, implying that the explanatory and response variables are linearly related.</p> Signup and view all the answers

What is the coefficient of determination for the relation between time between eruptions and length of eruption?

<p>88.0%</p> Signup and view all the answers

Study Notes

Univariate vs Bivariate Data

  • Univariate data involves the measurement of a single variable for each individual.
  • Bivariate data entails measuring two variables for each individual.

Associations Between Variables

  • Positive association indicates that as one variable increases, the other variable also increases.
  • Negative association describes a scenario where an increase in one variable leads to a decrease in the other variable.

Correlation and Causation

  • Correlation does not imply causation; a properly designed experiment is necessary to establish a causal relationship.
  • Observational study data cannot conclusively indicate causality.

Scatter Diagrams and Linear Relationships

  • A scatter diagram represents the relationship between two quantitative variables.
  • Data points that do not align in a straight line do not exhibit a linear relationship.
  • Linear relations can be classified as positive or negative based on the direction of the trend in the scatter diagram.

Linear Correlation Coefficient

  • The linear correlation coefficient (r) ranges from -1 to +1, where:
    • r = +1 indicates a perfect positive linear relationship.
    • r = -1 indicates a perfect negative linear relationship.
    • r close to 0 suggests little or no linear relation.
  • Examples of correlation coefficients:
    • r = 0.787 suggests a strong positive relationship.
    • r = -0.946 suggests a strong negative relationship.
    • r = 0.049 indicates almost no relationship.

Residuals

  • A residual is the difference between the observed value of a response variable and its predicted value.
  • Positive residuals imply that observed values exceed predicted values.

Least-Squares Regression Line

  • Points on the least-squares regression line represent predicted y-values corresponding to x-values based on the data.

Correlation vs Regression Analysis

  • The slope of the regression line corresponds with the correlation:
    • A negative correlation results in a negative slope.

Practical Examples of Correlation

  • Example of negative correlation: outside temperature and the number of people wearing coats.
  • Example of positive correlation: the number of doctors and administrators at a hospital.
  • Example of no correlation: size of the TV in a living room and the heating bill.

Lurking Variables

  • A lurking variable can influence the relationship between two variables, such as the economy affecting both cell phone prevalence and crime rates.### Scatter Diagrams and Regression Analysis
  • The x-axis of the scatter diagram ranges from 0 to 6, and the y-axis is also from 0 to 6.
  • Six points plotted: (1,5), (2,5.8), (4,4.8), (5,3.4), (5,3), (6,2.8).
  • The plotted points suggest a negative association, indicating that as x increases, y tends to decrease.
  • The least-squares regression line formula is given by (y = b_1x + b_0), where:
    • (b_1) is the slope calculated as (b_1 = r \cdot \frac{sy}{sx}).
    • (b_0) is the y-intercept given by (b_0 = \bar{y} - b_1\bar{x}).

Calculation of Regression Line

  • Slope (b_1 = -0.571).
  • Using (b_1) to calculate the intercept (b_0):
    • (b_0 = 4.0500 - (-0.571)(3.8333) = 6.239).
  • The least-squares regression line is (y = -0.571x + 6.239).
  • The line indicates a downward trajectory, represented graphically alongside the scatter diagram.

Pediatric Study on Height and Head Circumference

  • A study investigates the relationship between height (x) and head circumference (y) of 11 children.
  • The slope of the regression line is (0.081), implying:
    • For every 1 inch increase in height, head circumference increases by approximately 0.081 inches.
  • The y-intercept, at (15.1), does not carry practical significance since height cannot be zero.

Predictions and Residuals

  • For a child with a height of 24 inches, predicted head circumference (= 17.04) inches.
  • If the observed head circumference of the child is 17.4 inches, the residual is calculated as:
    • (residual = observed - predicted = 17.4 - 17.04 = 0.36).
  • This positive residual indicates that the observed value is above the predicted value from the regression model.

Scatter and Residual Plots

  • A scatter plot with the regression line and labeled residuals enhances understanding of data discrepancies.
  • Variations in head circumference among children of the same height can arise from biological differences, measurement errors, or other factors.

Coefficient of Determination

  • Defined as (R^2), measures the proportion of total variation in the response variable explained by the regression line.
  • Total deviation can be expressed as:
    • (Total deviation = unexplained deviation + explained deviation).

Residual Plots

  • A residual plot visually assesses the adequacy of the linear model, displaying residuals on the vertical axis against the explanatory variable on the horizontal axis.
  • A well-fitted linear model will show a random distribution of residuals, while patterns indicate potential issues in linearity or variance constancy.

Summary of Analysis Techniques

  • Techniques used in statistical analysis involve plotting data to visualize relationships, calculating regression lines, and interpreting slopes and intercepts.
  • Errors such as residuals provide insight into the accuracy of predictions made by the regression model, aiding in refining models and understanding underlying relationships.### Residual Plots and Linear Models
  • A U-shaped pattern in a residual plot indicates a violation of linear model assumptions.
  • Constant error variance is when the spread of residuals remains steady as the explanatory variable increases.
  • Outliers are extreme observations that deviate from the data's overall pattern; they can be identified in a residual plot where residuals lie far from others.
  • Absence of outliers in a residual plot suggests a potential linear relationship between explanatory and response variables.

Analyzing Scatter Diagrams

  • Coefficient of determination (R²) quantifies the variation in the response variable explained by the regression line.
  • Values indicate strength of correlation:
    • R² close to 1 indicates strong association.
    • R² = 0.58, 0.94, and 0.01 represent weak to strong correlations in different scatter diagrams.
  • A scatter diagram with all points on a line indicates perfect correlation (R² = 1).

Influence of Data Points

  • Influential points can significantly impact the least-squares regression line and correlate with a notable change in the slope or intercept.
  • Analysis of whether a point is influential can be guided by comparing regression lines with and without the outlier.

Geyser Eruptions Case Study

  • Data indicates a positive linear association between time between geyser eruptions and eruption length.
  • The residual plot validates the linear relationship, as no patterns discernible imply good model appropriateness.
  • The coefficient of determination (88.4%) suggests that the least squares regression line accounts for a substantial portion of variation in eruption length.

Planetary Sidereal Year and Distance

  • A scatter diagram illustrates the relationship between distance from a star and the sidereal year of a planet.
  • The linear correlation coefficient (r = 0.987) shows a strong positive relationship, indicating a significant linear correlation.
  • A least-squares regression line is computed to further analyze the relationship: y = 0.0624x - 12.2.
  • A residual plot is generated to verify the quality of the regression model with respect to the explanatory variable.

Summary of Key Concepts

  • Residual analysis is crucial for checking linear model validity.
  • Coefficient of determination (R²) is essential for quantifying explained variation.
  • Influential data points require careful examination in regression analysis.
  • Strong correlation coefficients indicate a high degree of linear dependence between variables.

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Explore the key concepts of marginal distributions found in contingency tables through this quiz. Test your understanding of how marginal distributions relate to frequency distributions and the interpretation of row and column variables. Perfect for students studying statistics.

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