7  Exercises 7

Exercise 7.1 A computer scientist has developed an algorithm to generate pseudo-random integers uniformly over the interval \([0,9]\). He codes the algorithm and generates \(1000\) pseudo-random numbers. Data on the frequency of occurrence of each digit from \(0\) to \(9\) are shown in the following table.

Generated number	Observed occurence, \(N_i\)	Observed frequency, \(O_i\)
0	94	0.094
1	93	0.093
2	112	0.112
3	101	0.101
4	104	0.104
5	95	0.095
6	100	0.100
7	99	0.099
8	108	0.108
9	94	0.094
\(n\)	1000	1.00

Does the random number generator works correctly at the \(5\%\) threshold?

Exercise 7.2 We wish to test the hypothesis that the number of defects on printed circuit boards follows a Poisson distribution. We collect a random sample of \(n=60\) printed circuit boards and observe the number of defects. The following data are obtained:

Defects	Observed frequency
0	32
1	15
2	9
3	4

Can we say that the number of defects follows a Poisson distribution? (Remark: the parameter \(\lambda\) of the Poisson distribution is to be estimated)

Exercise 7.3 A company has to choose one of three pension plans. Management wants to know if the preference for a plan is independent of job category at the \(5\%\) threshold. The following table shows the opinions of a sample of \(500\) employees.

	\(X_2 =\) Pension Plan
\(X_1 =\) Job categories	1	2	3	Total
1: Executive employees	160	140	40	340
2: Employees paid by hour	40	60	60	160
Total	200	200	100	500

Test the independence between job category and plan preference.