Python 计算置信区间

Question

I'm trying to compute 10,000 upper and lower bound confidence intervals using the following script:

import numpy as np
import statistics as stat
from matplotlib import pyplot as plt

N = 10000
sigma = 1
mu = 10
n = 10

X = []
for i in range (N):
    X.append(mu + sigma*(np.random.normal(size=n)))
    Xbar = np.mean(X, axis=0)
    lower_CI = Xbar - 1.96*sigma/np.sqrt(n)
    upper_CI = Xbar + 1.96*sigma/np.sqrt(n)

After computing the intervals I need to find the fraction of intervals that include mu = 10. However, I only get 10 upper and lower bound intervals, not 10,000. Also, Xbar has 10 values. Why does it not have only one value since it is the mean of X?

Where am I going wrong on this?

Answer 1

First of all, the following is happening in the code you posted: mu + sigma*(np.random.normal(size=n)) gives you an array of n=10 samples of a the normal distribution with your mu and sigma. X.append(mu + sigma*(np.random.normal(size=n))) adds this to your X list, so that X becomes a list of arrays. Xbar = np.mean(X, axis=0) With axis=0, you tell numpy to calculate the mean along axis 0 of your list of arrays. This means calculating a mean for each index of the arrays in your list of arrays X which is why Xbar has 10 entries. lower_CI = Xbar - 1.96*sigma/np.sqrt(n) here you set lower_CI to Xbar minus some sigma based number for your confidence interval. Because Xbar has 10 entries, the result will also have 10 entries. However, because you overwrite lower_CI in every iteration of the for loop instead of storing the values in a list, you do not get 10,000 lower bounds.

I am not completely sure what you are trying to do exactly, but the following code would approximate lower and upper confidence bounds for mu and sigma 10,000 times.

lower_CIs = []
upper_CIs = []
for i in range (N):
    X = mu + sigma*(np.random.normal(size=n))
    Xbar = np.mean(X)
    lower_CI = Xbar - 1.96*sigma
    upper_CI = Xbar + 1.96*sigma
    lower_CIs.append(lower_CI)
    upper_CIs.append(upper_CI)