In [1]:
from datascience import *
import numpy as np

%matplotlib inline
import matplotlib.pyplot as plots
plots.style.use('fivethirtyeight')

The GSI's Defense¶

In [2]:
scores = Table.read_table('scores_by_section.csv')
scores
Out[2]:
Section Midterm
1 22
2 12
2 23
2 14
1 20
3 25
4 19
1 24
5 8
6 14

... (349 rows omitted)

In [3]:
scores.group('Section')
Out[3]:
Section count
1 32
2 32
3 27
4 30
5 33
6 32
7 24
8 29
9 30
10 34

... (2 rows omitted)

In [4]:
scores.group('Section', np.average).show()
Section Midterm average
1 15.5938
2 15.125
3 13.6667
4 14.7667
5 17.4545
6 15.0312
7 16.625
8 16.3103
9 14.5667
10 15.2353
11 15.8077
12 15.7333
In [5]:
observed_average = 13.6667
In [6]:
random_sample = scores.sample(27, with_replacement=False)
random_sample
Out[6]:
Section Midterm
6 17
11 19
2 5
8 20
4 11
11 13
8 19
6 16
5 24
12 23

... (17 rows omitted)

In [7]:
np.average(random_sample.column('Midterm'))
Out[7]:
17.333333333333332
In [8]:
# Simulate one value of the test statistic 
# under the hypothesis that the section is like a random sample from the class

def random_sample_midterm_avg():
    random_sample = scores.sample(27, with_replacement = False)
    return np.average(random_sample.column('Midterm'))
In [9]:
# Simulate 50,000 copies of the test statistic

sample_averages = make_array()

for i in np.arange(50000):
    sample_averages = np.append(sample_averages, random_sample_midterm_avg())
In [10]:
# Compare the simulated distribution of the statistic
# and the actual observed statistic
averages_tbl = Table().with_column('Random Sample Average', sample_averages)
averages_tbl.hist(bins = 20)
plots.scatter(observed_average, -0.01, color='red', s=120);

Conventions About Inconsistency¶

Approach 1¶

In [11]:
# (1) Calculate the p-value: simulation area beyond observed value
np.count_nonzero(sample_averages <= observed_average) / 50000
# (2) See if this is less than 5%
Out[11]:
0.05612

Approach 2¶

In [12]:
# (1) Find simulated value corresponding to 5% of 50,000 = 2500
five_percent_point = averages_tbl.sort(0).column(0).item(2500)
five_percent_point
Out[12]:
13.62962962962963
In [13]:
# (2) See if this value is greater than observed value
observed_average
Out[13]:
13.6667

Visual Representation¶

In [14]:
averages_tbl.hist(bins = 20)
plots.plot([five_percent_point, five_percent_point], [0, 0.35], color='gold', lw=2)
plots.title('Area to the left of the gold line: 5%');
plots.scatter(observed_average, -0.01, color='red', s=120);