Chapter 26, Problem 10E

To determine

To perform goodness of fit test.

Answer to Problem 10E

There is not sufficient evidence to reject the claim that all digits show up with equal frequency.

Explanation of Solution

Given:

Digit	Observed
zero	99959
one	99758
two	100026
three	100229
four	100230
five	100359
six	99548
seven	99800
eight	99985
nine	100106

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:p_1=p_2=\mathrm{...}=p_{10}=\frac{1}{10}=0.1\\ H_a:H_a:Atleastoneofthe{p}_i’sdifferent\end{array}$

Using excel,

Observed	expected	O - E	(O - E)² / E	% of chisq
99959	100000.000	-41.000	0.017	0.31
99758	100000.000	-242.000	0.586	10.63
100026	100000.000	26.000	0.007	0.12
100229	100000.000	229.000	0.524	9.52
100230	100000.000	230.000	0.529	9.60
100359	100000.000	359.000	1.289	23.39
99548	100000.000	-452.000	2.043	37.08
99800	100000.000	-200.000	0.400	7.26
99985	100000.000	-15.000	0.002	0.04
100106	100000.000	106.000	0.112	2.04
1000000	1000000.000	0.000	5.509	100.00
5.51	chi-square
9	df
.7879	p-value

  ${\chi }^2$ = 5.51 and P-value = 0.7879

Decision: The p-value > 0.05, fail to reject H0.

Conclusion: There is not sufficient evidence to reject the claim that all digits shows up with equal frequency.