def t_test(data1, data2=None, tail='both', mu=0, equal=True):
assert tail in ['both', 'left',
'right'], 'tail should be one of "both","left","right"'
if data2 is None:
mean_val = mean(data1)
se = std(data1)/sqrt(len(data1))
t_val = (mean_val-mu)/se
df = len(data1)-1
else:
n1 = len(data1)
n2 = len(data2)
mean_diff = mean(data1)-mean(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
if equal:
sw = sqrt(((n1-1)*sample1_var+(n2-1)*sample2_var)/(n1+n2-2))
t_val = (mean_diff-mu)/(sw*sqrt(1/n1+1/n2))
df = n1 + n2 - 2
else:
se = sqrt(sample1_var/n1+sample2_var/n2)
t_val = (mean_diff-mu)/se
df_numerator = (sample1_var/n1+sample2_var/n2)**2
df_denominator = (sample1_var/n1)**2/(n1-1) + \
(sample2_var/n2)**2/(n2-1)
df = df_numerator/df_denominator
if tail == "both":
p = 2*(1-t.cdf(abs(t_val), df))
elif tail == "left":
p = t.cdf(t_val, df)
else:
p = 1-t.cdf(t_val, df)
return t_val, df, p
def mean_one_sided_lower_ci_est(data,alpha,sigma = None):
n = len(data)
if sigma is None: # 未知总体方差，使用t分布
t_value = abs(t.ppf(alpha,n-1))
s = std(data)
return mean(data) - s / sqrt(n) * t_value,inf
else: # 知道总体方差，使用标准正态分布
z_value = abs(norm.ppf(alpha))
return mean(data) - sigma/sqrt(n) * z_value,inf
def mean_ci_est(data, alpha, sigma=None): # confidence interval
n = len(data)
sample_mean = mean(data)
if sigma is None:
s = std(data)
se = s/sqrt(n)
t_value = abs(t.ppf(alpha/2,n-1))
return sample_mean - se * t_value, sample_mean + se * t_value
else:
se = sigma/sqrt(n)
z_value = abs(norm.ppf(alpha / 2)) # ppf默认下分位点，故使用abs
return sample_mean - se * z_value, sample_mean + se * z_value
def mean_ci_est(data, alpha, sigma=None):
n = len(data)
sample_mean = mean(data)
if sigma is None:
# 方差未知
s = std(data)
se = s/sqrt(n)
t_value = abs(t.ppf(alpha/2, n-1))
return sample_mean - se * t_value, sample_mean + se * t_value
else:
# 方差已知
se = sigma/sqrt(n)
z_value = abs(norm.ppf(alpha/2))
return sample_mean - se * z_value, sample_mean + se * z_value
def mean_ci_est(data, alpha, sigma=None):
"""均值的区间估计"""
n = len(data)
sample_mean = mean(data)
if sigma is None:
# 方差未知
s = std(data)
me = s / np.sqrt(n)
t_value = abs(t.ppf(alpha/2, n-1))
return round(sample_mean - me * t_value, 2), round(sample_mean + me * t_value, 2)
else:
# 方差已知
me = sigma / np.sqrt(n)
z_value = abs(norm.ppf(alpha/2))
return round(sample_mean - me * z_value, 2), round(sample_mean + me * z_value, 2)
def t_test(data1, data2=None, tail='both', mu=0.0, equal=True):
assert tail in ['both', 'left',
'right'], 'tail should be one of "both", "left", "right"'
if data2 is None:
# 单个总体的情况
mean_val = mean(data1)
se = std(data1) / np.sqrt(len(data1))
t_val = (mean_val - mu) / se
df = len(data1) - 1
else:
# 两个总体的情况
n1 = len(data1)
n2 = len(data2)
mean_diff = mean(data1) - mean(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
if equal:
# 方差相等的情况
sw = np.sqrt((((n1 - 1) * sample1_var + (n2 - 1) * sample2_var)) /
(n1 + n2 - 2))
t_val = (mean_diff - mu) / (sw * np.sqrt(1 / n2 + 1 / n2))
df = n1 + n2 - 2
else:
# 方差不等的情况
se = np.sqrt(sample1_var / n1 + sample2_var / n2)
t_val = (mean_diff - mu) / se
df = (sample1_var / n1 + sample2_var / n2)**2 / (
(sample1_var / n1)**2 / (n1 - 1) + (sample2_var / n2)**2 /
(n2 - 1))
if tail == 'both':
# 双尾检验
p = 2 * (1 - t.cdf(abs(t_val), df))
elif tail == 'left':
# 左尾检验
p = t.cdf(t_val, df)
else:
# 右尾检验
p = 1 - t.cdf(t_val, df)
return round(t_val, 2), round(df, 2), p
def mean_ci_est(data, alpha, sigma=None):
"""
总体方差未知，求均值的置信空间
总体方差已知，求均值的置信空间
data为传入的样本; alpha,sigma为需要传入的置信水平,sigma的值
"""
n = len(data) #求样本容量
sample_mean = mean(data) #求样本均值
if sigma is None: #方差未知
s = std(data) #求样本方差
se = s / sqrt(n) #求标准误
t_value = abs(t.ppf(alpha / 2, n - 1)) #求Z
return sample_mean - se * t_value, sample_mean + se * t_value
else: #方差已知
se = sigma / sqrt #求标准误
z_value = abs(norm.ppf(alpha /
2)) #求Z,由于取的Z alpha/2默认是返回坐标左边的面积，所以需要取绝对值
return sample_mean - se * z_value, sample_mean + se * z_value
# 测试频率
print(frequency(data))
# 测试众数
print(mode(data))
# 测试中位数
print(median(data))
# 测试均值
print(mean(data))
# 测试极差
print(rng(data))
# 测试四分位数
print(quartile(data))
# 测试方差
print(variance(data))
# 测试标准差
print(std(data))
总体均值未知，求方差的置信空间
data为传入的样本; alpha为需要传入的置信水平的值
"""
n = len(data) #求样本容量
s2 = variance(data) #求样本方差
chi2_lower_value = chi2.ppf(
alpha / 2, n - 1) #求坐标左侧Z面积，没错你没看错，因为数学证明的过程中是以右侧为基准的，但是scipy是以左侧为基准的
chi2_upper_value = chi2.ppf(1 - alpha / 2, n - 1) #求坐标右侧Z面积
return (n - 1) * s2 / chi2_upper_value, (n - 1) * s2 / chi2_lower_value
if __name__ == '__main__':
salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113,
1601] #18岁月收入数据
salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345,
2182] #35岁月收入数据
print(mean(salary_18)) #平均月收入的点估计
print(mean_ci_est(salary_18, 0.05)) #平均月收入的区间估计
print(mean(salary_35)) #平均月收入的点估计
print(mean_ci_est(salary_35, 0.05)) #平均月收入的区间估计
print()
print(std(salary_18)) #整体方差的点估计开根
print(variance(salary_18)) #整体方差的点估计(样本方差)
print(var_ci_est(salary_18, 0.05)) #区间估计
print(std(salary_35)) #整体方差的点估计开根
print(variance(salary_35)) #整体方差的点估计(样本方差)
print(var_ci_est(salary_35, 0.05)) #区间估计
print()