Question

3. You also have been asked to determine at both the 2% and 5% levels of significance whether the proportion of supporters of the candidate that describes itself as generally loyal to the political party of the candidate has decreased recently from its previously hypothesized value of 80%. You select a random sample of supporters of the candidate for study. The sample data concerning whether the supporters generally describe themselves as party loyalists is shown below. Answer the question as to whether there has been a decrease in the population proportion of supporters that describes itself as party loyalists from its previous value of 80%.

Appendix Two: Party Loyalist? (Y = yes, N = no)

Y         Y         Y         N         Y         N         Y         N         Y         N         Y

N         Y         Y         Y         Y         Y         N         N         N         Y         Y

Y         Y         Y         Y         Y         Y         Y         Y         Y         Y         Y

N         Y         N         N         Y         N         Y         Y         N         Y         N

Y         N         N         N         Y         Y         Y         N         Y         Y         N

Answer 1

Consider population proportion P

P: Population proportion of supporters that describes itself as party loyalist.

p : sample proportion of supporters that describes itself as party loyalist.

Number of yes = 36 and number of No = 19

p = 36 / ( 36+19) = 0.6545

We have to test the hypothesis that

Whether or not there has decrease in the Population proportion of supporters that describes itself as party loyalist from its previous value of 80%?

i.e. Null Hypothesis -

against

Alternative Hypothesis - ( left-tailed test)

We use one sample z-test for testing population proportion.

The value of test statistic is

Value of test statistic Z = -2.6994

Since the test is left-tailed and value of test statistic is -2.6994. p-value is obtained by

p-value = P ( Z < -2.6994)

from normal probability table

P ( Z < - 2.6994) = 0.0038

p-value = 0.0038.

Since p-value is less than 0.02.

Hence both 2% and 5% level of significance we reject the null hypothesis.

Conclusion: There is sufficient evidence support to claim that there has decrease in the Population proportion of supporters that describes itself as party loyalist from its previous value of 80%.