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.

# 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)

def one_bootstrap_mean():
    resample = births.sample()
    return np.average(resample.column('Maternal Age'))

# 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())

# 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

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

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.

# Try estimating it from the sample

estimated_SD_of_sample_average = sample_SD / (sample_size**0.5)
estimated_SD_of_sample_average

0.16972373742299796

# Approximate 95% confidence interval for population mean

sample_average - 2*estimated_SD_of_sample_average, sample_average + 2*estimated_SD_of_sample_average

(26.888831911866099, 27.567726861558093)

SD of 0/1 population

# 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

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)

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

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

zero_one_pop.iscatter('Proportion of Ones', 'Pop SD')