Question

When public opinion surveys are conducted by mail, a cover letter explaining the purpose of the survey is usually included. To determine whether the wording of the cover letter influences the response rate, three different cover letters were used in a survey of students at a Midwestern university. Suppose that each of the three cover letters accompanied questionnaires sent to an equal number of randomly selected students. Returned questionnaires were then classified according to the type of cover letter (I, II, or III). Use the accompanying data to test the hypothesis that ?₁ = 1/3, ?₂ = 1/3, and ?₃ = 1/3, where ?₁, ?₂ and ?₃ are the true proportions of all returned questionnaires accompanied by cover letters I, II, and III, respectively. Use a 0.05 significance level. (Use 2 decimal places.)

	Cover-letter Type
	I	II	III
Frequency	51	36	31

?² =  

Answer 1

The null and alternative hypothesis is

H0: ?1 = ?2 = ?3 = 1/3

H1: At least one proportion is not equal.

Level of significance = 0.05

Test statistic is

O: Observed frequency
E: Expected frequency.

E = n*pi

	O	p	E	(O-E)	(O-E)^2	(O-E)^2/E
	51	0.333333	39.33333	11.66667	136.1111	3.460452
	36	0.333333	39.33333	-3.33333	11.11111	0.282486
	31	0.333333	39.33333	-8.33333	69.44444	1.765537
Total	118					5.51

Degrees of freedom = Number of E's - 1 = 3 - 1 = 2

Critical value = 5.991

Test statistic < critical value we fail to reject null hypothesis.

Conclusion:

?1 = ?2 = ?3 = 1/3