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经典例题

某校高二年级共有四个班，采用四种不同的教学方法进行数学教学，为了比较这四种教学法的效果是否存在明显的差异，期末统考后，从这四个班中各抽取 5 名考生的成绩，如下所示。

问这四种教学法的效果是否存在显著性差异（α =0.05）？

1.计算F值

import numpy as np
from scipy.stats import f_oneway# Data for the four classes
class1 = [75, 77, 70, 88, 72, 80, 79, 81]
class2 = [93, 80, 85, 90, 84, 86, 85, 81]
class3 = [65, 67, 77, 68, 65, 64, 62, 68]
class4 = [72, 70, 71, 65, 81, 72, 68, 74]# Perform one-way ANOVA
f_statistic, p_value = f_oneway(class1, class2, class3, class4)# Output the results
print("F-statistic:", f_statistic)
print("P-value:", p_value)# Interpret the results
alpha = 0.05
if p_value < alpha:print("There is a significant difference in the effectiveness of the teaching methods.")
else:print("There is no significant difference in the effectiveness of the teaching methods.")

F-statistic: 22.045992451864645
P-value: 1.5622062333927252e-07
There is a significant difference in the effectiveness of the teaching methods.

2.计算SS、df和F值

import numpy as np
import pandas as pd
from scipy.stats import f_oneway, f# Data for the four classes
class1 = [75, 77, 70, 88, 72, 80, 79, 81]
class2 = [93, 80, 85, 90, 84, 86, 85, 81]
class3 = [65, 67, 77, 68, 65, 64, 62, 68]
class4 = [72, 70, 71, 65, 81, 72, 68, 74]# Perform one-way ANOVA
f_statistic, p_value = f_oneway(class1, class2, class3, class4)# Degrees of freedom
num_groups = 4
num_samples = len(class1) + len(class2) + len(class3) + len(class4)
df_between = num_groups - 1
df_within = num_samples - num_groups# Calculate sum of squares (SS)
mean_total = np.mean([np.mean(class1), np.mean(class2), np.mean(class3), np.mean(class4)])
ss_total = np.sum((np.concatenate([class1, class2, class3, class4]) - mean_total) ** 2)
ss_between = np.sum([len(class1) * (np.mean(class1) - mean_total) ** 2,len(class2) * (np.mean(class2) - mean_total) ** 2,len(class3) * (np.mean(class3) - mean_total) ** 2,len(class4) * (np.mean(class4) - mean_total) ** 2])
ss_within = np.sum((class1 - np.mean(class1)) ** 2) + \np.sum((class2 - np.mean(class2)) ** 2) + \np.sum((class3 - np.mean(class3)) ** 2) + \np.sum((class4 - np.mean(class4)) ** 2)# Calculate mean squares (MS)
ms_between = ss_between / df_between
ms_within = ss_within / df_within# Calculate F-statistic
f_statistic_manual = ms_between / ms_within# Critical F-value
alpha = 0.05
f_crit = f.ppf(1 - alpha, df_between, df_within)# Create a DataFrame for better tabular representation
data = {'Class 1': class1,'Class 2': class2,'Class 3': class3,'Class 4': class4,
}df = pd.DataFrame(data)# Output the ANOVA results
print("Analysis of Variance (ANOVA):")
print("F-statistic (from scipy.stats):", f_statistic)
print("P-value (from scipy.stats):", p_value)
print("\nManual Calculation:")
print("SS Between:", ss_between)
print("SS Within:", ss_within)
print("DF Between:", df_between)
print("DF Within:", df_within)
print("MS Between:", ms_between)
print("MS Within:", ms_within)
print("F-statistic (manual calculation):", f_statistic_manual)
print("Critical F-value:", f_crit)# Interpret the results
if p_value < alpha:print("\nThere is a significant difference in the effectiveness of the teaching methods.")
else:print("\nThere is no significant difference in the effectiveness of the teaching methods.")

Manual Calculation:
SS Between: 1538.59375
SS Within: 651.375
DF Between: 3
DF Within: 28
MS Between: 512.8645833333334
MS Within: 23.263392857142858
F-statistic (manual calculation): 22.045992451864645
Critical F-value: 2.9466852660172655