Homework 10: Regression Inference¶

Helpful Resource:

• Python Reference: Cheat sheet of helpful array & table methods used in Data 8!

Recommended Reading:

• Using Confidence Intervals

• The Regression Line

• Inference for Regression

Please complete this notebook by filling in the cells provided. Before you begin, execute the cell below to setup the notebook by importing some helpful libraries. Each time you start your server, you will need to execute this cell again.

Throughout this homework and all future ones, please be sure to not re-assign variables throughout the notebook! For example, if you use max_temperature in your answer to one question, do not reassign it later on. Otherwise, you will fail tests that you thought you were passing previously!

Deadline:

This assignment is due Wednesday, 4/22 at 11:00am PT. Submissions after this time will be accepted for 24 hours and will incur a 20% penalty. Any submissions later than this 24 hour period will not be accepted unless an extension has been granted as per the syllabus page. Turn it in by Tuesday, 4/21 at 11:00am PT for 5 extra credit points.

Directly sharing answers is not okay, but discussing problems with the course staff or with other students is encouraged. Refer to the syllabus page to learn more about how to learn cooperatively.

You should start early so that you have time to get help if you’re stuck. Office hours are held Monday through Friday in Warren Hall 101B or online. The office hours schedule appears here.

The point breakdown for this assignment is given in the table below:

0. Data 8 Final Exam Accomodations Form¶

You can find the final exam accomodations form here. All students must fill out the form so we can better accomodate everyone for the final exam.

Question 0.1. Fill out the final exam accomodations form linked above. Once you have submitted, a secret word will be displayed. Set secret_word to the secret string at the end of the form. (6 points)

An Introduction to Regression Inference¶

Previously in this class, we’ve used confidence intervals to quantify uncertainty about estimates. We can also run hypothesis tests using a confidence interval under the following procedure:

1. Define a null and alternative hypothesis (they must be of the form “The parameter is X” and “The parameter is not X”).

2. Choose a p-value cutoff, and call it $q$, where $q$ is a percentage.

3. Construct a $(100\%-q)\%$ confidence interval using bootstrap sampling (for example, if your p-value cutoff is 1%, or 0.01, then construct a 99% confidence interval).

4. Using the confidence interval, determine if your data are more consistent with your null or alternative hypothesis, at your $q\%$ p-value cutoff:

   • If the null hypothesis parameter X is in your confidence interval, the data are more consistent with the null hypothesis.

   • If the null hypothesis parameter X is not in your confidence interval, the data are more consistent with the alternative hypothesis.

More recently, we’ve discussed the use of linear regression to make predictions based on correlated variables. For example, we can predict the height of children based on the heights of their parents.

We can combine these two topics to make powerful statements about our population by using the following techniques:

• Bootstrapped interval for the true slope

• Bootstrapped prediction interval for y (given a particular value of x)

This homework explores these two methods.

The Data¶

The Snowy Plover is a tiny bird that lives on the coast in parts of California and elsewhere. It is so small that it is vulnerable to many predators, including people and dogs that don’t look where they are stepping when they go to the beach. It is considered endangered in many parts of the U.S.

The data are about the eggs and newly-hatched chicks of the Snowy Plover. Here’s a picture of a parent bird incubating its eggs.

The data were collected at the Point Reyes National Seashore by a former student at Berkeley. The goal was to see how the size of an egg could be used to predict the weight of the resulting chick. The bigger the newly-hatched chick, the more likely it is to survive.

Each row of the table below corresponds to one Snowy Plover egg and the resulting chick. Note how tiny the bird is:

• Egg Length and Egg Breadth (widest diameter) are measured in millimeters

• Egg Weight and Bird Weight are measured in grams; for comparison, a standard paper clip weighs about one gram

In this investigation, we will be using the egg weight to predict bird weight. Run the cell below to create a scatter plot of the egg weights and bird weights, along with their line of best fit.

1. Finding the Bootstrap Confidence Interval for the True Slope¶

Looking at the scatter plot of our sample, we observe a linear relationship between egg weight and bird weight. However, relationships that we have identified in the sample might not be representative of the true relationship in the population.

We want to know if there truly exists a linear relationship between egg weight and bird weight for Snowy Plovers. If there is no linear relationship between the two variables, then we’d expect a correlation of 0. Consequently, the slope of the regression line would also be 0.

We highly recommend reviewing section 16.2 before attempting this part of the homework

Question 1.1. Let’s run a hypothesis test using confidence intervals to see if there is a linear relationship between egg weight and bird weight. Choose the appropriate null and alternative hypotheses that will allow you to conduct this test. Assign the numbers corresponding to the correct null and alternative hypotheses to null_hypothesis and alt_hypothesis. (12 points)

1. The true slope of the regression line, computed using our sample of Snowy Plovers, is 0.

2. The true slope of the regression line, computed using our sample of Snowy Plovers, is not 0.

3. The true slope of the regression line, computed using our sample of Snowy Plovers, is greater than 0.

4. The true slope of the regression line, computed using the population of all Snowy Plovers, is 0.

5. The true slope of the regression line, computed using the population of all Snowy Plovers, is not 0.

6. The true slope of the regression line, computed using the population of Snowy Plovers, is greater than 0.

Hint: Our regression line predicts bird weight from egg weight!

Question 1.2. Define the following two functions:

1. standard_units: This function takes in an array of numbers and returns an array containing those numbers converted to standard units.

2. correlation: This function takes in a table and two column names (one for x and one for y) and returns the correlation between these columns.

(4 points)

Question 1.3. Using the functions you just implemented, create a function called fit_line. It should take a table (e.g. birds) and the column names associated to x and y as its arguments and return an array containing the slope and intercept of the regression line (in that order) that predicts the y column in the table using the x column. (8 points)

Run this cell to plot the line produced by calling fit_line on the birds table.

Note: You are not responsible for the code in the cell below, but make sure that your fit_line function generated a reasonable line for the data.

Now we have all the tools we need to create a confidence interval that quantifies our uncertainty about the true relationship between egg weight and bird weight.

Question 1.4. Create an array called resampled_slopes that contains the slope of the best fit line for 1000 bootstrap resamples of birds. Plot the distribution of these slopes. (8 points)

Hint: Use the fit_line function you defined in 1.3.

Question 1.5. Use your resampled slopes to construct an 95% confidence interval for the true value of the slope. (8 points)

Question 1.6. Based on your confidence interval, would you reject or fail to reject the null hypothesis that the true slope is 0? Why? What p-value cutoff are you using?

Select all statements that are correct given your 95% confidence interval from the previous question, and assign your answers to the array ci_conclusion. (10 points)

1. Because 0 is outside of our 95% confidence interval, there is evidence that the true slope is non-zero.

2. Because 0 is outside of our 95% confidence interval, there is evidence the true slope is zero.

3. We are using a p-value cutoff of 0.05.

4. We are using a p-value cutoff of 0.95.

5. Based on our confidence interval and p-value cutoff, we would fail to reject the null hypothesis.

6. Based on our confidence interval and p-value cutoff, we would reject the null hypothesis.

Hint: Read the introduction of this homework!

Question 1.7. Assume that the 95% confidence interval you generated in Question 1.5 was $[0.6, 0.8]$.

Select all statements that are correct interpretations of this interval, and assign your answers to the array confidence_interval_uses. (8 points)

1. This interval allows us to make a conclusion about whether the true slope is 0, at a 5% p-value cutoff.

2. There is a 95% chance that the true slope falls between $[0.6, 0.8]$.

3. We are 95% confident that the true slope is somewhere between $[0.6, 0.8]$.

2. Finding the Bootstrap Prediction Interval¶

Suppose we’re visiting Point Reyes and stumble upon some Snowy Plover eggs; we’d like to know how heavy they’ll be once they hatch. In other words, we want to use our regression line to make predictions about a bird’s weight based on the weight of the corresponding egg.

However, just as we’re uncertain about the slope of the true regression line, we’re also uncertain about the predictions made based on the true regression line.

Question 2.1. Define the function fitted_value. It should take in four arguments:

1. table: a table like birds. We’ll be predicting the values in the second column using the first.

2. x_col: the name of our x-column within the input table

3. y_col: the name of our y-column within the input table

4. given_x: a number, the value of the predictor variable for which we’d like to make a prediction.

The function should return the line’s prediction for the given x. (6 points)

Hint: Make sure to use the fit_line function you defined in Question 1.3.

Question 2.2. Raymond, the resident Snowy Plover expert at Point Reyes, tells us that the egg he has been carefully observing has a weight of 9 grams. Using fitted_value above, assign the variable experts_egg as the predicted bird weight for Raymond’s egg. (4 points)

A fellow parkgoer raises the following objection to your prediction:

“Your prediction depends on your sample of 44 birds. Wouldn’t your prediction change if you had a different sample of 44 birds?”

Having read section 16.3 of the textbook, you know just the response! Had the sample been different, the regression line would have been different too. This would ultimately result in a different prediction. To see how good our prediction is, we must get a sense of how variable the prediction can be.

Question 2.3. Define a function compute_resampled_line that takes in a table tbland two column names, x_col and y_col, and returns an array containing the parameters of the best fit line (slope and intercept) for one bootstrapped resample of the table. (6 points)

Run the following cell below in order to define the function bootstrap_lines. It takes in four arguments:

1. tbl: a table like birds

2. x_col: the name of our x-column within the input tbl

3. y_col: the name of our y-column within the input tbl

4. num_bootstraps: an integer, a number of bootstraps to run.

It returns a table with one row for each bootstrap resample and the following two columns:

1. Slope: the bootstrapped slopes

2. Intercept: the corresponding bootstrapped intercepts

Question 2.4. Create an array called predictions_for_eight that contains the predicted bird weights based on an egg of weight 8 grams for each regression line in regression_lines. (8 points)

Question 2.5. Create an approximate 95% confidence interval for these predictions. (6 points)

Question 2.6. Set plover_statements to an array of integer(s) that correspond to statement(s) that are true. (6 points)

1. The 95% confidence interval covers 95% of the bird weights for eggs that had a weight of eight grams in birds.

2. The 95% confidence interval gives a sense of how much actual weights differ from your prediction.

3. The 95% confidence interval quantifies the uncertainty in our estimate of what the true line would predict.

You’re all done with Homework 10!

Important submission steps:

1. Run the tests and verify that they all pass.

2. Choose Save Notebook from the File menu, then run the final cell.

3. Click the link to download the zip file.

4. Go to Pensive and submit the zip file to the corresponding assignment. The name of this assignment is “HW 10 Autograder”.

It is your responsibility to make sure your work is saved before running the last cell.

Pets of Data 8¶

Fluffy hopes you have a wonderful rest of your week!

Congrats on finishing Homework 10!

To double-check your work, the cell below will rerun all of the autograder tests.

Submission¶

Make sure you have run all cells in your notebook in order before running the cell below, so that all images/graphs appear in the output. The cell below will generate a zip file for you to submit. Please save before exporting!