中心极限定理 & 区间估计

import scipy.stats
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

matplotlib inline这句，可以不用plt.show出图

​

标准正态分布

standard_norm = scipy.stats.norm

x = np.arange(-4, 4, 0.01)
plt.plot(x, standard_norm.pdf(x))
plt.show()

错误记录：
拼写，standard，我少了一个a，错写为：standrd。 一直没找到错误原因

output_3_0.png

t 分布

t_dist = scipy.stats.t

plt.plot(x, standard_norm.pdf(x), label='standard normal')

x = np.arange(-4, 4, 0.01)
plt.plot(x, t_dist.pdf(x, df=1), label='t distribution')

plt.legend()
plt.show()

错误记录：
plt.plot(x,y,label='abc')，中的参数label，被我写成了labels复数。

output_5_0.png

置信区间

$$ \bar{x} - |z_{\alpha/2}|\frac{\sigma}{\sqrt{n}} < \mu < \bar{x} + |z_{\alpha/2}|\frac{\sigma}{\sqrt{n}}$$

导入住房面积数据

house = pd.read_csv('house_size.csv', header=None)

house_size = house.iloc[:,0]

print(list(house_size))

314, 119, 217, 326, 342, 318, 130, 465, 383, 396, 507, 283, 250, 326, 279, 363, 229, 303, 367, 246,

247, 262, 209, 294, 112, 249, 354, 355, 272, 277, 377, 411, 223, 232, 445, 333, 336, 349, 611, 516, 233, 275, 395, 241, 127, 228, 305, 321, 235, 226, 288, 503, 305, 280, 318, 281, 227, 279, 171, 290, 336, 284, 380, 314, 316, 476, 309, 293, 160, 300, 319, 396, 275, 212, 344, 305, 280, 331, 359, 283, 136, 322, 359, 202, 188, 187, 457, 340, 262, 288, 318, 381, 289, 205, 373, 200, 320, 213, 261, 357]

假设已知总体的标准差为 86 平方米

pop_std = 86

sample_mean = house_size.mean()
sample_mean

300.85

sample_size = len(house_size)
sample_size

100

z_score = scipy.stats.norm.isf(0.025)  # 95% 置信度
z_score

1.9599639845400545

margin_error = z_score * pop_std / np.sqrt(sample_size)
margin_error

16.855690267044469

c_limit = sample_mean - margin_error
upper_limit = sample_mean + margin_error

print('95%% Confidence Interval: ( %.1f, %.1f)' % (lower_limit, upper_limit))

95% Confidence Interval: ( 284.0, 317.7)

定义置信区间计算的函数

def ci_z(data, pop_std, confidence):
    sample_mean = np.mean(data)
    sample_size = len(data)
    
    alpha = (1 - confidence) / 2
    z_score = scipy.stats.norm.isf(alpha)
    
    ME = z_score * pop_std / np.sqrt(sample_size)

    lower_limit = sample_mean - ME
    upper_limit = sample_mean + ME
    
    return (lower_limit, upper_limit)

ci_z(house_size, pop_std, 0.90)

(286.70425880821733, 314.99574119178271)

ci_z(house_size, pop_std, 0.95)

(283.99430973295557, 317.70569026704447)

ci_z(house_size, pop_std, 0.99)

(278.69786798947951, 323.00213201052054)

ci_z(house_size, pop_std, 1)

(-inf, inf)

错误记录1,sqrt，是表示二次开方， 需要些np.sqrt(data)

错误记录2： z——score在取值的时候，已经把置信度放进去了，isf是指两段大于0.025的部分。 所以我在此处就不需要再乘以α/2了

Bootstrap 方法计算置信区间

np.random.choice(house_size, size=10)  # 从house_size数据中随机抽取10个数据，可重复抽取

array([326, 226, 212, 226, 279, 294, 373, 383, 226, 377])

def bootstrap_mean(data):
    # 从数据data中重复抽样，样本大小与data相同，并返回样本均值
    return np.mean(np.random.choice(data, size=len(data)))

def draw_bootstrap(data, times=1):
    
    #初始化长度为times的空数组
    bs_mean = np.empty(times)
    
    #进行多次（times次）抽样，将每次得到的样本均值存储在bs_mean中
    for i in range(times):
        bs_mean[i] = bootstrap_mean(data)
        
    return bs_mean

bs_mean = draw_bootstrap(house_size, 10000)
plt.hist(bs_mean, bins=50, normed=True, rwidth=0.9)
plt.show()

output_29_0.png

np.percentile(bs_mean, [2.5, 97.5])

array([ 283.659,  318.66 ])

作业

用t分布求上述置信区间

import scipy.stats
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

%config InlineBackend.figure_format = 'retina'

house = pd.read_csv('house_size.csv', header=None)
house_size = house.iloc[:,0]
print(house_size.head())

0    314
1    119
2    217
3    326
4    342
Name: 0, dtype: int64

t_dist = scipy.stats.t
print(t_dist)

<scipy.stats._continuous_distns.t_gen object at 0x7f5e05844f98>

t_score = scipy.stats.t(df=1).isf(0.025)
print(t_score)

12.7062047364

sample_mean = house_size.mean()
sample_size = len(house_size)
sample_std = np.std(house_size)
print(sample_mean)
print(sample_size)

300.85
100

t_score = scipy.stats.t(df=sample_mean-1).isf(0.025)
print(t_score)

1.96790699855

ME = t_score*sample_std/np.sqrt(sample_size)
print(ME)
#错误记录S是样本标准差，我竟然使用使用的是样本均值去计算。

17.4183977393

lower_limit = sample_mean - ME
upper_limit = sample_mean + ME
print(lower_limit,upper_limit)

283.431602261 318.268397739

来吧，试一下用函数呢

def ci_t(data,confidengce):
    sample_size = len(data)
    sample_std = np.std(data)
    sample_mean = np.mean(data)
    t_score = scipy.stats.t(df=sample_size-1).isf((1-confidengce)/2)
    ME = t_score*sample_std/np.sqrt(sample_std)
    l_limit = sample_mean-ME
    u_limit = sample_mean+ME
    return(l_limit,u_limit)

ci_t(house_size,0.95)

(282.18229316004903, 319.51770683995102)

ci_t(house_size,0.00001)

(300.84988178885078, 300.85011821114927)