Summarizing Numerical Associations

STAT 20: Introduction to Probability and Statistics

Warmup Questions

Correlation

If the correlation coefficient for a given scatterplot is \(r = -0.81\), which of the following must be true about the relationship between the explanatory and response variable?

1. The association has a linear form.
2. The association must be positive.
3. The association must be negative.
4. The association is weakened by outliers.

• Part (c) The association must be negative.

Simple Linear Model

In a study, volunteers kept a record of all the food they ate. The amount of protein (in grams per week) and the amount of fat (in grams per week) was then calculated for each person. Here is a summary of the entire data set:

Fat (g/wk)	Protein (g/wk)
mean = 668	mean = 662
sd = 65	sd = 70

Correlation(fat, protein) = 0.8

Calculate the equation of the least-squares regression line to predict protein in terms of fat.

• \(b_1 = (0.8) \times \frac{70}{65} \approx 0.86\)
• \(b_0 = 662 - (0.86)(668) \approx 87.52\)
• \(\hat{y} = 87.52 + 0.86 x\)

A pediatric clinic models the relationship between a child’s age in months (x) and their average weight in kilograms (y). Using data from thousands of healthy toddlers, they calculate the line of best fit:

predicted y = 0.4x + 7.2

Provide a verbal interpretation for the slope (0.4)

For every additional month of age (+1 month on the x-axis), a toddler’s weight is predicted to increase by an average of 0.4 kg (y-axis)

Concept Questions

Take a guess:

Which state has the highest poverty rate? The lowest?

This can be asked without poll-everywhere, without group discussion, just asking students to shout out which states they think will have highest and lowest.

As they answer, ask where they’re thinking those states points are; how they’re using this plot to think about the highest poverty rate and the lowest poverty rate. Point out that these two questions don’t even take advantage of the scatter plot, which allows you to see the relationship between two variables. Each question just deals with the marginal distribution of poverty or graduation rate, which could just as well be two separate histograms, as shown on the next slide.

Marginal distributions of poverty and graduation rate

This is an equally-useful set of plots to answer the previous question.

If we want to ask a question that requires a scatter plot (and a linear model) to answer, we need to ask a conditional question.

The scatter plot and linear model answer “conditional” questions, such as:

• Which state is most outperforming the model’s expectations?
• That is, which state has the highest graduation rate given its poverty rate?

01:00

This assesses whether or not students have an informal sense of what a residual is.

• Which state is most outperforming the model’s expectations?
• That is, which state has the highest graduation rate given its poverty rate?

• Which state is most underperforming the model’s expectations?
• That is, which state has the lowest graduation rate given its poverty rate ?

• Which state is most underperforming the model’s expectations?
• That is, which state has the lowest graduation rate given its poverty rate ?

Worksheet: Summarizing Numerical Associations

20:00

Break

05:00

Lab: Flights II

25:00