In [1]:
import matplotlib
from datascience import *
%matplotlib inline
import matplotlib.pyplot as plots
import numpy as np
plots.style.use('fivethirtyeight')

Lecture 28¶

Review: Lecture 24¶

From a random sample, construct a 95% confidence interval for the ages of the mothers in the population.

In [2]:
# original sample

births = Table.read_table('baby.csv')
births.show(3)
Birth Weight Gestational Days Maternal Age Maternal Height Maternal Pregnancy Weight Maternal Smoker
120 284 27 62 100 False
113 282 33 64 135 False
128 279 28 64 115 True

... (1171 rows omitted)

In [3]:
def one_bootstrap_mean():
    resample = births.sample()
    return np.average(resample.column('Maternal Age'))
In [4]:
# Generate means from 3000 bootstrap samples
num_repetitions = 3000
bstrap_means = make_array()
for i in np.arange(num_repetitions):
    bstrap_means = np.append(bstrap_means, one_bootstrap_mean())
In [5]:
# Get the endpoints of the 95% confidence interval
left = percentile(2.5, bstrap_means)
right = percentile(97.5, bstrap_means)

print(f"Left End: {left}")
print(f"Right End: {right}")

Left End: 26.906303236797275
Right End: 27.579216354344123
In [6]:
resampled_means = Table().with_columns(
    'Bootstrap Sample Mean', bstrap_means
)
resampled_means.hist(bins=15)
plots.plot([left, right], [0, 0], color='yellow', lw=8, label='95% CI')
plots.legend();

Method Based on CLT¶

In [7]:
sampled_ages = births.column('Maternal Age')
sample_size = len(sampled_ages)
sample_average = np.average(sampled_ages)
sample_SD = np.std(sampled_ages)

print(f"Sample Size: {sample_size}")
print(f"Sample Average: {sample_average}")
print(f"Sample SD: {sample_SD}")

Sample Size: 1174
Sample Average: 27.228279386712096
Sample SD: 5.815360404190897

We need to add and subtract $2 \cdot ( \frac{\text{Population SD}}{\sqrt{\text{Sample_Size}}})$ but we don't have the population SD.

In [8]:
# Try estimating it from the sample

estimated_SD_of_sample_average = sample_SD / (sample_size**0.5)
estimated_SD_of_sample_average
Out[8]:
0.16972373742299796
In [9]:
# Approximate 95% confidence interval for population mean

sample_average - 2*estimated_SD_of_sample_average, sample_average + 2*estimated_SD_of_sample_average
Out[9]:
(26.888831911866099, 27.567726861558093)

SD of 0/1 population¶

In [10]:
# population of size 10

number_of_ones = 4
zero_one_population = np.append(np.ones(number_of_ones), np.zeros(10 - number_of_ones))

print(f"Population: {zero_one_population}")
print(f"Standard Deviation: {np.round(np.std(zero_one_population),2)}")

Population: [ 1.  1.  1.  1.  0.  0.  0.  0.  0.  0.]
Standard Deviation: 0.49
In [11]:
def sd_of_zero_one_population(number_of_ones):
    """Returns the SD of a population 
    that has 10 elements: num_ones with value 1 and (10 - num_ones) with value 0"""
    zero_one_population = np.append(np.ones(number_of_ones), 
                                    np.zeros(10 - number_of_ones))
    return np.std(zero_one_population)
In [12]:
possible_ones = np.arange(11)
zero_one_pop = Table().with_columns(
    'Number of Ones', possible_ones,
    'Proportion of Ones', possible_ones / 10
)
zero_one_pop.show()
Number of Ones Proportion of Ones
0 0
1 0.1
2 0.2
3 0.3
4 0.4
5 0.5
6 0.6
7 0.7
8 0.8
9 0.9
10 1
In [13]:
sds = zero_one_pop.apply(sd_of_zero_one_population, 'Number of Ones')
zero_one_pop = zero_one_pop.with_column('Pop SD', sds)
zero_one_pop.show()
Number of Ones Proportion of Ones Pop SD
0 0 0
1 0.1 0.3
2 0.2 0.4
3 0.3 0.458258
4 0.4 0.489898
5 0.5 0.5
6 0.6 0.489898
7 0.7 0.458258
8 0.8 0.4
9 0.9 0.3
10 1 0
In [14]:
zero_one_pop.iscatter('Proportion of Ones', 'Pop SD')
In [ ]: