Question

1.

Consider a multinomial experiment with n = 245 and k = 4. The null hypothesis to be tested is H₀: p₁ = p₂ = p₃ = p₄ = 0.25. The observed frequencies resulting from the experiment are: (You may find it useful to reference the appropriate table: chi-square table or F table)

Category	1	2	3	4
Frequency	72	45	60	68

Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

2.

The quality department at an electronics company has noted that, historically, 94% of the units of a specific product pass a test operation, 4% fail the test but are able to be repaired, and 2% fail the test and need to be scrapped. Due to recent process improvements, the quality department would like to test if the rates have changed. A recent sample of 500 parts revealed that 478 parts passed the test, 16 parts failed the test but were repairable, and 6 parts failed the test and were scrapped. (You may find it useful to reference the appropriate table: chi-square table or F table)

Compute the value of the test statistic. (Round the intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

Answer 1

1) applying chi square test statistic:

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.250	72.000	61.25	1.37	1.887
2	0.250	45.000	61.25	-2.08	4.311
3	0.250	60	61.25	-0.16	0.026
4	0.250	68	61.25	0.86	0.744
total	1.000	245	245		6.967

value of the test statistic =6.967

2)

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.940	478.000	470.00	0.37	0.136
2	0.040	16.000	20.00	-0.89	0.800
3	0.020	6	10.00	-1.26	1.600
total	1.000	500	500		2.536

value of the test statistic =2.536