Lab 5: Simulations¶

Welcome to Lab 5! The data used in this lab will contain salary data and other statistics for basketball players from the 2023-2024 NBA season. This data was collected from the following sports analytic sites: Basketball Reference and Hoops Hype.

Recommended Reading:

• Iteration

• Simulations

• Randomness

• Python Reference

Getting help on lab: Whenever you feel stuck or need some further clarification, find a GSI or tutor, and they’ll be happy to help!

As a reminder, here are the policies for getting full credit if you are in regular lab (Lab is worth 20% of your final grade):

• 80% of lab credit will be attendance-based. To receive attendance credit for lab, you must attend the full discussion portion (first hour) at which point the GSI will take attendance.

• The remaining 20% of credit will be awarded for submitting the programming-based assignment to Pensive by the deadline (5pm on Friday) with all test cases passing.

If you are in self-service lab, 100% of credit will be awarded for submitting the programming-based assignment to Pensive by the deadline (5pm on Friday) with all test cases passing.

Submission: Once you’re finished, run all cells besides the last one, select File > Save Notebook, and then execute the final cell. The result will contain a zip file that you can use to submit on Pensive.

Let’s begin by setting up the tests and imports by running the cell below.

1. Nachos and Conditionals¶

In Python, the boolean is a data type with only two possible values: True and False. Expressions containing comparison operators such as < (less than), > (greater than), and == (equal to) evaluate to Boolean values. A list of common comparison operators can be found below!

Run the cell below to see an example of a comparison operator in action.

We can even assign the result of a comparison operation to a variable. Note that == and = are not the same!

Just like arithmetic operators can be applied on every item of an array, comparison operators can also be used on arrays to compare an entire array with some value. The output of this comparison is an array of boolean values.

One day, when you come home after a long week, you see a hot bowl of nachos waiting on the dining table! Let’s say that whenever you take a nacho from the bowl, it will either have only cheese, only salsa, both cheese and salsa, or neither cheese nor salsa (a sad tortilla chip indeed).

Let’s try and simulate taking nachos from the bowl at random using the function, np.random.choice(...).

np.random.choice¶

np.random.choice picks one item at random from the given array. It is equally likely to pick any of the items. Run the cell below several times, and observe how the results change. Tip: To keep running a cell multiple times you can use the keyboard shortcut ctrl + return.

To repeat this process multiple times, pass in an int n as the second argument to return n different random choices. By default, np.random.choice samples with replacement. If you pass in an int n, it will return an array of items. Sampling with replacement means that after an element is drawn, it is replaced back to where you are sampling from and can be drawn again in the future. If we sample n times with replacement, each time, every element has an equal chance of being selected.

Run the next cell to see an example of sampling with replacement 10 times from the nachos array.

To count the number of times a certain type of nacho is randomly chosen, we can use np.count_nonzero.

np.count_nonzero¶

np.count_nonzero counts the number of non-zero values that appear in an array. When an array of boolean values are passed through the function, it will count the number of True values (remember that in Python, True is coded as 1 and False is coded as 0.)

Run the next cell to see an example that uses np.count_nonzero.

Question 1.1 Assume we took ten nachos at random, and stored the results in an array called ten_nachos as done below. Find the number of nachos with only cheese using code (do not manually enter the final answer).

Hint: Our solution involves a comparison operator (e.g. ==, <, ...) and the np.count_nonzero method.

Conditional Statements

A conditional statement is a multi-line statement that allows Python to choose among different alternatives based on the truth value of an expression.

Here is a basic example.

def sign(x):
    if x > 0:
        return 'Positive'
    else:
        return 'Negative'

If the input x is greater than 0, we return the string 'Positive'. Otherwise, we return 'Negative'.

If we want to test multiple conditions at once, we use the following general format.

if <if expression>:
    <if body>
elif <elif expression 0>:
    <elif body 0>
elif <elif expression 1>:
    <elif body 1>
...
else:
    <else body>

Only the body for the first conditional expression that is true will be evaluated. Each if and elif expression is evaluated and considered in order, starting at the top. elif can only be used if an if clause precedes it. As soon as a true value is found, the corresponding body is executed, and the rest of the conditional statement is skipped. If none of the if or elif expressions are true, then the else body is executed.

For more examples and explanation, refer to the section on conditional statements here.

Question 1.2 Complete the following conditional statement so that the string 'More please' is assigned to the variable say_please if the number of nachos with cheese in ten_nachos is less than 5. Use the if statement to do this (do not directly reassign the variable say_please).

Hint: You should be using number_cheese from Question 1.

Question 1.3 Write a function called nacho_reaction that returns a reaction (as a string) based on the type of nacho passed in as an argument. Use the table below to match the nacho type to the appropriate reaction.

Hint: If you’re failing the test, double check the spelling of your reactions.

Question 1.4 Create a table ten_nachos_reactions that consists of the nachos in ten_nachos as well as the reactions for each of those nachos. The columns should be called Nachos and Reactions.

Hint: Consider using the apply method, which returns an array.

Question 1.5 Using code, find the number of ‘Wow!’ reactions for the nachos in ten_nachos_reactions.

2. Simulations and For Loops¶

Using a for statement, we can perform a task multiple times. This is known as iteration. The general structure of a for loop is:

for <placeholder> in <array>: followed by indented lines of code that are repeated for each element of the array being iterated over. You can read more about for loops here.

NOTE: We often use i as the placeholder in our class examples, but you could name it anything! Some examples can be found below.

One use of iteration is to loop through a set of values. For instance, we can print out all of the colors of the rainbow.

We can see that the indented part of the for loop, known as the body, is executed once for each item in rainbow. The name color is assigned to the next value in rainbow at the start of each iteration. Note that the name color is arbitrary; we could easily have named it something else. Whichever name we pick, we need to use it consistently throughout the for loop.

In general, however, we would like the variable name to be somewhat informative.

Furthermore, we can count how many colors are in the rainbow by using a variable that is incremented by 1 with every iteration of the loop!

num_colors = 0
for color in rainbow:
    num_colors = num_colors + 1

Because there are 7 items in the rainbow array, the loop will run 7 times, adding 1 to num_colors each time. After our for loop, num_colors will be assigned to 7

Question 2.1 In the following cell, we’ve loaded the text of Pride and Prejudice by Jane Austen, split it into individual words, and stored these words in an array p_and_p_words. Using a for loop, assign longer_than_five to the number of words in the novel that are more than 5 letters long.

Hint: You can find the number of letters in a word with the len function.

Hint: How can you use longer_than_five to keep track of the number of words that are more than five letters long?

Another way we can use for loops is to repeat lines of code many times. Recall the structure of a for loop:

for <placeholder> in <array>: followed by indented lines of code that are repeated for each element of the array being iterated over.

Sometimes, we don’t care about what the value of the placeholder is. We instead take advantage of the fact that the for loop will repeat as many times as the length of our array. In the following cell, we iterate through an array of length 5 and print out “Hello, world!” in each iteration, but we don’t need to use the placeholder i in the body of our for loop.

Question 2.2 Using a simulation with 10,000 trials, assign num_different to the number of times, in 10,000 trials, that two words picked uniformly at random (with replacement) from Pride and Prejudice have different lengths.

Hint 1: How did we use this function in section 1 to sample at random (with replacement) from an array?

Hint 2: Remember that != checks for non-equality between two items.

3. Sampling Basketball Data¶

We will now introduce the topic of sampling, which we’ll be discussing in more depth in this week’s lectures. We’ll guide you through this code, but if you wish to read more about different kinds of samples before attempting this question, you can check out section 10 of the textbook.

Run the cell below to load full_data which contains NBA player and salary data that we will use for our sampling.

Imagine that we had gotten data on only a smaller subset of the players. For 525 players, it’s not so unreasonable to expect to see all the data, but usually we aren’t so lucky.

If we want to make estimates about a certain numerical property of the population, we may have to come up with these estimates based only on a smaller sample. The numerical property of the population is known as a parameter, and the estimate is known as a statistic (e.g. the mean or median). Whether these estimates are useful or not often depends on how the sample was gathered. We have prepared some example sample datasets to see how they compare to the full NBA dataset. Later we’ll ask you to create your own samples to see how they behave.

To save typing and increase the clarity of your code, we will package the analysis code into a few functions. This will be useful in the rest of the lab as we will repeatedly need to create histograms and collect summary statistics from that data.

We’ve defined the histograms function below, which takes a table with columns Age and Salary and draws a histogram for each one. It uses bin widths of 1 year for Age and $1,000,000 for Salary.

Question 3.1. Create a function called compute_statistics that takes a table containing an “Age” column and a “Salary” column and:

• Draws histograms of ages and salaries

• Returns a two-element array containing the average age and average salary (in that order)

You can call the histograms function to draw both histograms at once!

Simple random sampling¶

A more justifiable approach is to sample uniformly at random from the players. In a simple random sample (SRS) without replacement, we ensure that each player is selected at most once. Imagine writing down each player’s name on a card, putting the cards in an box, and shuffling the box. Then, pull out cards one by one and set them aside, stopping when the specified sample size is reached.

Producing simple random samples¶

Sometimes, it’s useful to take random samples even when we have the data for the whole population. It helps us understand sampling accuracy.

sample¶

The table method sample produces a random sample from the table. By default, it draws at random with replacement from the rows of a table. Sampling with replacement means for any row selected randomly, there is a chance it can be selected again if we sample multiple times. Sample takes in the sample size as its argument and returns a table with only the rows that were selected. This differs from np.random.choice, which takes an array and outputs a random value from the array.

Run the cell below to see an example call to sample() with a sample size of 5, with replacement.

The optional argument with_replacement=False can be passed through sample() to specify that the sample should be drawn without replacement.

Run the cell below to see an example call to sample() with a sample size of 5, without replacement.

Question 3.2 Produce a simple random sample without replacement of size 44 from full_data , and assign it to my_small_srswor_data in the cell below. Then, run your analysis on it again by using the compute_statistics function you defined above. Run the cell a few times to see how the histograms and statistics change across different samples.

• How much does the average age change across samples?

• What about average salary?

(FYI: srs = simple random sample, wor = without replacement)

Type your answer here, replacing this text.

4. More Random Sampling Practice¶

More practice for random sampling using np.random.choice.

Simulations and For Loops (cont.)¶

Question 4.1 We can use np.random.choice to simulate multiple trials.

Bing and Mia decide to play a game together rolling a standard six-sided die. Their score on each roll is determined by the face that is rolled. They want to know what their total score would be if they rolled the die 1000 times. Write code that simulates their total score after 1000 rolls.

Hint: First decide the possible values you can take in the experiment (point values in this case). Then use np.random.choice to simulate Bing and Mia’s rolls. Finally, sum up the rolls to get the total score.

Simple random sampling (cont.)¶

Question 4.2 As in the previous question, analyze several simple random samples of size 100 from full_data by using the compute_statistics function.

• Do the histogram shapes seem to change more or less across samples of 100 than across samples of size 44?

• Are the sample averages and histograms closer to their true values/shape for age or for salary? What did you expect to see?

Type your answer here, replacing this text.

Fubo, Perry, and Dino are very happy that you finished the lab! Congratulations!

You’re done with lab!

Important submission information:

• Run all the tests and verify that they all pass

• Save from the File menu

• Run the final cell to generate the zip file

• Click the link to download the zip file

• Then, go to Pensive and submit the zip file to the corresponding assignment. The name of this assignment is “Lab XX Autograder”, where XX is the lab number -- 01, 02, 03, etc.

• If you finish early in Regular Lab, ask one of the staff members to check you off.

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

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!