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More on relationships between two variables Unit 5-3 Power models Relationships between categorical variables. Review of common transformations for achieving linearity. A problem dealing with area might benefit from squaring the data (power of 2) since area involves square units.
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More on relationships between two variables Unit 5-3 Power models Relationships between categorical variables
Review of common transformationsfor achieving linearity • A problem dealing with area might benefit from squaring the data (power of 2) since area involves square units. • A problem dealing with weight or volume might benefit from cubing or cube-rooting (a power of 3 or one-third) the data since volume involves cubic units. • Data involving a ratio (like miles per gallon) might benefit from a reciprocal transformation (power of -1). • Exponential data benefits from taking the logarithm of the response variables. (we learned this last time) • Power models may benefit from taking the logarithm of both variables.
y = axp Power law models When do we see power models? Area Volume Abundance of species
The theory behind logarithms makes it so taking the logarithm of both variables in a power model yields a linear relationship between logx and logy. Power Law Models y = axp
Notice the power p in the power law becomes the slope of the straight line that links logx to logy. Power Law Models y = axp yields logy= loga + plogx We can even roughly estimate what power p the law involves by finding the LSRL of logy on logx and using the slope of the line as an estimate of the power.
Body and brain weight of 96 species of mammals • You might remember this example from last time.
When we plot the logarithm of brain weight against the logarithm of body weight for all 96 species we get a fairly linear form. This suggests that a power law governs this relationship.
Bigfoot is estimated to weigh about 280 pounds or 127 kilograms. Use the model to predict Bigfoot's brain weight. Prediction from a power model
Predict Xena's period of revolution from the data if it is 9.5 billion miles from the sun. What's a planet anyway?
Five key questions What's a planet anyway?
Graphs Scatterplot Linearized plot residual plot What's a planet anyway?
Is it an exponential or power model? Linearize the data accordingly What's a planet anyway?
Numerical summaries What's a planet anyway?
Model Give an equation for our linear model with a statement of how well it fits the data What's a planet anyway?
Interpretation Is our model sufficient for making predictions? Predict Xena's period of revolution (an astronomical unit is 93 million miles) What's a planet anyway?
College Students • Two way table • Row variable • Column variable
College Students • Marginal distributions • Round off error
College Students • percents • Calculate the marginal distribution of age group in percents.
College Students • Each marginal distribution from a two-way table is a distribution for a single categorical variable. • Construct a bar graph that displays the distribution of age for college students.
College Students • To describe relationships among or compare categorical variables, calculate appropriate percents.
College Students • Conditional distributions • Compare the percents of women in each age group by examining the conditional distributions. • Find the conditional distribution of gender, given that a student is18 to 24 years old.
