In [1]:

    

alpha <- 0.05
n <- 10000
rate <- c(968, 1026, 1021, 975, 1012, 1046, 1021, 969, 948, 1014)
df <- data.frame(Ziffer=0:9, Häufigkeit=rate)


    

In [2]:

    

nc <- 10
p_hat = 1/nc
n_p = n * p_hat
chisq_comp = (df$Häufigkeit - n_p)^2 / n_p
df <- data.frame(df, p=p_hat, n_p=n_p, chisq_comp=chisq_comp)
df


    

    
Out[2]:





Ziffer
Häufigkeit
p
n_p
chisq_comp

	
1
0
968
0.1
1000
1.024
	
2
1
1026
0.1
1000
0.676
	
3
2
1021
0.1
1000
0.441
	
4
3
975
0.1
1000
0.625
	
5
4
1012
0.1
1000
0.144
	
6
5
1046
0.1
1000
2.116
	
7
6
1021
0.1
1000
0.441
	
8
7
969
0.1
1000
0.961
	
9
8
948
0.1
1000
2.704
	
10
9
1014
0.1
1000
0.196

In [3]:

    

Q <- sum(chisq_comp)
Q


    

    
Out[3]:




9.328

In [4]:

    

dof <- nc-1
chi <- qchisq(1-alpha, dof)
chi


    

    
Out[4]:




16.9189776046205

In [5]:

    

pi <- scan("pi10000.txt")


    

In [6]:

    

chisq.test(table(pi), p=rep(0.1, 10))


    

    
Out[6]:





	Chi-squared test for given probabilities

data:  table(pi)
X-squared = 9.328, df = 9, p-value = 0.4076