Question

A politician claims that he will gain 50% of votes in a city election and thereby he will win. However, after studying the politician's policies, an expert thinks he will lose the election; i.e. he will gain less than 50% of the votes. To test his conjecture, the expert randomly selected n voters from the city and found Y of them would vote for the candidate. Let p denotes the supporting rate for the candidate.

α)State clearly the null and alternative hypotheses for the above research question.

β)Νow suppose that n = 60 and for the test in α) your rejection region is Y ≤ 20. Which is the asymptotic distribution of Y, under the null hypothesis, due to the Central Limit Theorem?

c)Using, for example, the command “=IF(RAND()<0,4;1;0)” in Excel, simulate 40 different realizations of the above experiment with n = 60 and the true supporting rate to be 40%. In order to “freeze” the output generated by Excel, select the cells with all 0s and 1s values that have been randomly generated using the above mentioned command, copy them and paste them (using the paste special option) as values back on top of the cells. Then in each of the 40 different simulated “datasets” calculate Y, i.e. total number of 1s (voters). Finally create a histogram of your 40 values of Y and discuss your findings

Answer 1

Answer:

Given that a politician claims that he will gain 50% of votes in a city election. An expert thinks he will lose the election.  Let p denotes the supporting rate for the candidate.

a)

null and alternative hypotheses :

b)

The asympolic distribution of Y, the null hypothesis is due to the central limit therom is normal distribution with mean = n_p=60 * 0.5

c)

n = 60 and the true supporting rate to be 40%.