Question

Two years ago, a political party ASR received 9:8% of the votes in an election. To study the current political preferences, a statistical research institute plans to organise a poll by the end of the present year. In this study, n voters will be interviewed about the political party they prefer. Below, p, denotes the proportion of voters that would vote ASR if the elections were held now. Furthermore, bp denotes the (random) sample proportion of the ASR voters.

(a) Suppose that n = 500. Determine an interval that with probability 0:90 will contain the (not yet observed) realisation of bp if p were the same as two years ago. (6)

(b) Find random bounds (depending on p) that will include the proportion p with probability 0:95. Express the width of the accompanying interval in terms of p and n. (4)

(c) The interval in part (b) will be the starting point to create, at the end of the current year, when the data are observed, an interval that will probably contain the proportion p. How large should the sample size be to obtain an interval width about 0:02? (Hint: Substitute the former proportion 0:098 for p in the width of part (b).) (5)

(d) At the end of the year, n randomly chosen voters are interviewed with n as calculated in part (c); the realisation of bp turns out to be 0:853. Use the interval in part (b) as a starting point to create an interval that will probably contain the population proportion p. Is the width of that interval indeed about 0:02? (5)

Answer 1

Two years ago, a political party ASR received 9:8% of the votes in an election. To study the current political preferences, a statistical research institute plans to organise a poll by the end of the present year.

In this study,

n = be the number voters in the political party they prefer.

p = be the proportion of voters that would vote ASR if the elections were held.

bp = be the (random) sample proportion of the ASR voters.

a )

We need to construct the 90% confidence interval for the population proportion. We have been provided with the following information about the sample proportion:

Sample proportion   = 0.098

Sample size N = 500

Formula for the confidence interval for the population proportion is given by,

Where,

= be the critical value for the alpha = 0.1

Therefore

The corresponding confidence interval is computed as shown below:

b )

Here we have to calculate width of the accompanying interval in terms of p and n. We are given the proportion p with probability 0.95

An formula for the interval width in terms of p is given by,

Where,

= be the critical value for the alpha = 0.05

Therefore

An formula for the interval width in terms of n  is given by,

N = 502.34

c )

How large should the sample size be to Obtain an interval width about 0:02?

An interval width is,  

Therefore,

N = 2392

d )

Therefore,

N = 713.