Question

A poll conducted by the Shire Institute for Statistical Research discovered that only 39% of 164 randomly sampled inhabitants of the West Farthing enjoy boating on the Brandywine River, while the same poll discovered that 53% of 143 randomly sampled inhabitants of Buckland enjoy this activity. Perform a significance test at α = 0.05 to determine if the proportion of Hobbits in the West Farthing who enjoy boating is significantly lower than the proportion of Hobbits in Buckland who do.

10.1 Provide the hypotheses for this test:

10.2 Provide the value for p̂_pooled:

10.3 Provide the value of the numerator of the test statistic:

10.4 Provide the value of the denominator of the test statistic:

10.5 Provide the final answer of the test statistic:

10.6 Provide the p-value:

10.7 Provide the comparison needed to make the conclusion:

10.8 Provide the conclusion for the test:

Answer 1

Two sample proportion test (z-test)

Aim: determine if the proportion of Hobbits in the West Farthing who enjoy boating is significantly lower than the proportion of Hobbits in Buckland who do.

Given: two samples : West Farthing and Buckland

Sample 1: sample size (n1)= 164

         Proportion (p1) = the proportion of Hobbits in the West Farthing who enjoy boating

         P1 =39% = 0.39

Sample 2: sample size (n2)=143

Proportion (p2)= the proportion of Hobbits in the Buckland who enjoy boating

P2= 53% = 0.53

10.1Hypothesis

Null Hypothesis (H0) : p1 = p2

        i.e. there is no significant difference between proportions of Hobbits the West Farthing and in    

        the Buckland who enjoy boating

       Alternative Hypothesis (H1) : p1 < p2

i.e. the proportion of Hobbits in the West Farthing who enjoy boating is significantly lower than the proportion of Hobbits in Buckland who do

10.2   p̂pooled =ppooled

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p= (0.39*164+0.53*143)/(164+143)

               p= 0.455212

10.3 the numerator of the test statistic:

         Numerator = p1-p2 =0.39-0.53= -0.14

10.4 the denominator of the test statistic:

SE = sqrt{ p * ( 1 - p ) * ( (1/n1) + (1/n2) ) }

SE= sqrt {0.4552*(1-0.4552)*((1/164)+(1/143)))

SE= 0.056977

10.5 the test statistic:

Test statistic z= (p1-p2)/SE= -2.45713

10.6 the p-value: 0.0070

(From EXCEL : =NORM.S.DIST(z,TRUE)

10.7 the conclusion: Here p-value = 0.007 < alpha =0.05. therefore we reject H0.

10.8 the conclusion for the test: the proportion of Hobbits in the West Farthing who enjoy boating is significantly lower than the proportion of Hobbits in Buckland who do