Question

X and Y want to predict the outcome of the elections in their congressional district. X has been travelling to all neighborhoods of the district to ask people’s opinions about their favorite candidate. X has interview 2000 people and has found that 40% of them support the democrat candidate. Y has created a website where people can express their political opinions and he has advertised his website by putting some posters near his house. Y has found that 4000 people has entered his website and only 30% of them have expressed support for the democrat candidate.

1. Construct two99%confidenceintervalsforthetrueproportionofdemocrats in the district, one for X and one for Y.

2. Which confidence interval has the largest range?

3. Whose poll would you trust the most? Is there any reason why you would trust X’s results more than Y’s results?

Answer 1

Solution :

The 99% confidence interval for the population proportion is given as follows :

Where, p̂ is sample proportion, q̂ = 1 - p̂, n is sample size and Z_(0.01/2) is critical z-value to construct 99% confidence interval.

For X :

p̂ = 0.40,  q̂ = 1 - 0.40 = 0.60 and n = 2000

Using Z-table we get, Z_(0.01/2) = 2.576

Hence, the 99% confidence interval for true proportion of democrats in the district is,

The 99% confidence interval for true proportion of democrats in the district is (0.37, 0.43).

For Y :

p̂ = 0.30,  q̂ = 1 - 0.30 = 0.70 and n = 4000

Using Z-table we get, Z_(0.01/2) = 2.576

Hence, the 99% confidence interval for true proportion of democrats in the district is,

The 99% confidence interval for true proportion of democrats in the district is (0.39, 0.41).

The confidence interval constructed for X's has the larger ranger.

I would trust the X's poll. I would trust X's result more because he has travelled to all the neighiborhoods of the district and collected the data. So his sample is truly representing the population. But Y has collected the data by making a website. So it might be that his data did not represents the population. It might be that people from some neighborhood are not included in the sample.