Question

An ethnographer working in the municipal center of Maxcanu, in Yucatan, Mexico, noticed that this town had a lot more houses built with traditional thatch roofs. On the basis of architectural style, this ethnographer developed the hypothesis that Maxcanu was more traditional than most other municipal centers in Yucatan. She decided to test this hypothesis using data on the languages spoken by people in Maxcanu. In more traditional towns, many people speak native languages. The data in the sample from Maxcanu show that 8 people spoke only Spanish, 7 spoke only native Yucatec, and 37 are bilingual. Published information from other rural municipal centers shows that 20% of the population usually speaks only Spanish, 5% of the population usually speaks only Yucatec, and 75% of the population is bilingual. Do the data from Maxcanu deviate from these published expectations to such an extreme as to suggest that Maxcanu is not a typical rural municipal center? Frame you answer in the six steps of hypothesis testing.

Answer 1

Step 1:

Null Hypothesis H0: The languages spoken by people in Maxcanu follows the distribution of published expectations.

Alternative Hypothesis H0: The languages spoken by people in Maxcanu does not follows the distribution of published expectations.

Step 2:

Since the test is applied to a single categorical variable from two or more different populations, we will use Chi-square test of homogeneity to test the hypothesis.

Step 3:

Significance level = 0.05

Degree of freedom = number of groups - 1 = 3 - 1 = 2

Critical value of Chi Square test statistic at df = 2 and 0.05 significance level is 5.99

We reject the null hypothesis H0 if the test statistic is greater than 5.99

Step 4:

Expected count for people in Maxcanu speaking Spanish,  Yucatec and bilingual are,

, Ei = n * pi

n = 8 + 7 + 37 = 52

E1 = 52 * 0.2 = 10.4
E2 = 52 * 0.05 = 2.6
E3 = 52 * 0.75 = 39

Chi-square test statistic = (Oi - Ei)^2/Ei
= (8 - 10.4)^2 / 10.4 + (7 - 2.6)^2 / 2.6 + (37 - 39)^2 / 39

= 8.1026

Step 5:

Since the test statistic is greater than the critical value, we reject null hypothesis H0.

Step 6:

We conclude that there is significant evidence that the languages spoken by people in Maxcanu does not follows the distribution of published expectations.

Thus, there is significant evidence that the data from Maxcanu deviate from these published expectations to such an extreme as to suggest that Maxcanu is not a typical rural municipal center