Question

When public opinion surveys are conducted by mail, a cover letter explaining the purpose of the...

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 =

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