Reading:
Please complete this notebook by filling in the cells provided.
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# Don't change this cell; just run it.
import numpy as np
from datascience import *
# These lines do some fancy plotting magic.
import matplotlib
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
import warnings
warnings.simplefilter('ignore', FutureWarning)
import scipy.stats
#import otter
#grader = otter.Notebook()
Yanay gets a lot of spam calls. An area code is defined to be a three digit number from 200-999 inclusive. In reality, many of these area codes are not in use, but for this question we'll simplify things and assume they all are. Throughout these questions, you should assume that Yanay's area code is 781.
Question 1. Assuming each area code is just as likely as any other, what's the probability that the area code of two back to back spam calls are 781?
prob_781 = ...
prob_781
Question 2. Rohan already knows that Yanay's area code is 781. Rohan randomly guesses the last 7 digits (0-9 inclusive) of his phone number. What's the probability that Rohan correctly guesses Yanay's number, assuming he’s equally likely to choose any digit?
Note: A phone number contains an area code and 7 additional digits, i.e. xxx-xxx-xxxx
prob_yanay_num = ...
prob_yanay_num
Yanay suspects that there's a higher chance that the spammers are using his area code (781) to trick him into thinking it's someone from his area calling him. Ashley thinks that this is not the case, and that spammers are just choosing area codes of the spam calls at random from all possible area codes (Remember, for this question we’re assuming the possible area codes are 200-999, inclusive). Yanay wants to test his claim using the 50 spam calls he received in the past month.
Here's a dataset of the area codes of the 50 spam calls he received in the past month.
# Just run this cell
spam = Table().read_table('spam.csv')
spam
Question 3. Define the null hypothesis and alternative hypothesis for this investigation.
Hint: Don’t forget that your null hypothesis should fully describe a probability model that we can use for simulation later.
$H_o: \pi = \frac{1}{800}$
H0: pi = 1/800
$H_a: \pi > \frac{1}{800}$
Ha: pi > 1/800
Below is the code to run a binomial test, along with the results.
num_781 = spam.where("Area Code", 781).num_rows
total_calls = spam.num_rows
pval = scipy.stats.binom_test(num_781, total_calls, 1/800, alternative = "greater")
print(f"Binomial Test Results: The p-value = {pval}")
Question 4. Suppose you use a p-value cutoff of 5%. What do you conclude from the hypothesis test? Why?
Write your answer here, replacing this text.
Suppose now that we want to re-run this hypothesis test using the simulation approach.
Question 5. Which of the following test statistics would be a reasonable choice to help differentiate between the two hypotheses?
Hint: For a refresher on choosing test statistics, check out the textbook section on Test Statistics.
Assign reasonable_test_statistics to an array of numbers corresponding to these test statistics.
reasonable_test_statistics = make_array(1,4)
For the rest of this question, suppose you decide to use the number of times you see the area code 781 in 50 spam calls as your test statistic.
Question 6.
Write a function called simulate that generates exactly one simulated value of your test statistic under the null hypothesis. It should take no arguments and simulate 50 area codes under the assumption that the result of each area is sampled from the range 200-999 inclusive with equal probability. Your function should return the number of times you saw the 781 area code in those 50 random spam calls.
possible_area_codes = np.arange(200,1000)
def simulate():
sample = np.random.choice(possible_area_codes, 50, replace=True)
return sum(sample == 781)
#return Table().with_column("Sample", sample).where("Sample", 781).num_rows
# Call your function to make sure it works
simulate()
Question 7. Generate 20,000 simulated values of the number of times you see the area code 781 in 50 random spam calls. Assign test_statistics_under_null to an array that stores the result of each of these trials.
Hint: Use the function you defined in Question 6.
test_statistics_under_null = make_array()
repetitions = 20000
for i in np.arange(repetitions):
test_statistics_under_null = np.append(test_statistics_under_null,
simulate())
test_statistics_under_null
Question 8. Using the results from Question 7, generate a histogram of the empirical distribution of the number of times you saw the area code 781 in your simulation. NOTE: Use the provided bins when making the histogram
bins = np.arange(0,5,1) # Use these provided bins
Table().with_column("Test Statistics Under Null",
test_statistics_under_null).hist(bins=bins)
Question 9. Compute an empirical p-value for this test.
# First calculate the observed value of the test statistic from the `spam` table.
observed_val = spam.where("Area Code", 781).num_rows
print(observed_val)
p_value = ...
p_value
Question 10. Suppose you use a P-value cutoff of 1%. What do you conclude from the hypothesis test? Why?
Write your answer here, replacing this text.
Yanay collects information about this month's spam calls. The table with_labels is a sampled table, where the Area Code Visited column contains either "Yes" or "No" which represents whether or not Yanay has visited the location of the area code. The Picked Up column is 1 if Yanay picked up and 0 if he did not pick up.
# Just run this cell
with_labels = Table().read_table("spam_picked_up.csv")
with_labels.pivot("Picked Up", "Area Code Visited")
Yanay is going to perform a test to see whether or not he is more likely to pick up a call from an area code he has visited. Specifically, his null hypothesis is that there is no difference in the distribution of calls he picked up between visited and not visited area codes, with any difference due to chance. His alternative hypothesis is that there is a difference between the two categories, specifically that he thinks that he is more likely to pick up if he has visited the area code. We are going to perform a $\chi^2$ test for independence.
# Chi-squared test for independence
import numpy as np
obs = np.array([[17,18],[4,11]]) # Put data into array structure
import scipy.stats
Results = scipy.stats.chi2_contingency(obs)
## Don't change anything below this line
print(obs)
print("Chi-Square = ", Results[0])
print("p-value = ", Results[1])
print("degrees of freedom = ", Results[2])
print("Expected Table = ")
print(Results[3])
print("Reminder: No zero cells, and not more than 20% with counts below 5 in Expected Table.")
Question 11 Using a level of significance of 0.05, compare the p-value from the above and interpret the results in context. Is he more likely to pick up the phone if it comes from an area code he has visited?
Write your answer here, replacing this text.
Researcher's have been studying whether elephants are afraid of bees. If they are, this knowledge could be applied to keep elephants from damaging crops or from wandering into areas where the elephants could get hurt, using a so-called beehive fence.
Click here for a related video
Using a dataset provided via the internet by some researchers in Gabon, this table was created, showing whether an elephant visited a site containing fruit trees Never (0 times), Sometimes (1-5 times) or Often (6-11 times) and whether a beehive was present during the visit.
| Never | Sometimes | Often | |
|---|---|---|---|
| Hives Absent | 50 | 16 | 4 |
| Hives Present | 755 | 77 | 8 |
We want to run a $\chi^2$ test for independence to see if there's a relationship here that might imply that elephants are deterred by the presence of beehives.
Question 12. What are the null and alternative hypotheses in this situation?
Write your answer here, replacing this text.
Null is that there is no relationship, elephants are just as likely to visit sites with bees as to not.
Alternative is that there is some relationship, elephants avoid bees.
Question 13. Copy and adapt the code used to run the $\chi^2$ test above so as to run the test to determine if this data supports the claim that elephants are deterred by bees.
# Chi-squared test for independence
import numpy as np
obs = ... # Put data into array structure
import scipy.stats
Results = scipy.stats.chi2_contingency(obs)
## Don't change anything below this line
print(obs)
print("Chi-Square = ", Results[0])
print("p-value = ", Results[1])
print("degrees of freedom = ", Results[2])
print("Expected Table = ")
print(Results[3])
print("Reminder: No zero cells, and not more than 20% with counts below 5 in Expected Table.")
Question 14. Using a p-value cut-off of 0.05, what conclusion do you draw from this data?
Write your answer here, replacing this text.
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