A recent Gallup Poll dealt with the proportion of Canadians who favoured the various Prime Ministers. It found that only Pierre Trudeau did quite well, of all Prime Ministers in recent memory, and that the proportion favouring Brian Mulroney is quite low. A newspaper article included the following statement: "Results are based on 1,037 interviews with Canadian adults conducted July 6 - 9, 2006. Gallup says a sample this size is accurate within 4 percentage points, 19 in 20 times". In fact, the precision of their estimate of the population proportion was not 4 percentage points. Find the closest precision below, when estimating the population proportion at 50%.
a) .02
b) .025
c) .030
d) .035
To find the closest precision when estimating the population proportion at 50%, we need to determine the appropriate margin of error. In the statement mentioned, Gallup claims that the sample size of 1,037 is accurate within 4 percentage points, 19 in 20 times.
To estimate the population proportion at 50%, we use the formula for margin of error:
Margin of Error = Z * √(p(1-p)/n)
Where:
Z represents the standard score for the desired level of confidence (19 in 20 times corresponds to a 95% confidence level)
p represents the estimated population proportion (50% in this case)
n represents the sample size (1,037 in this case)
Using these values, we can calculate the margin of error:
Margin of Error = 1.96 * √((0.5(1-0.5))/1,037) = 1.96 * √(0.25/1,037) ≈ 0.028
Therefore, the closest precision to the claimed accuracy of 4 percentage points for estimating the population proportion at 50% is 0.03 (option c).
To find the closest precision for estimating the population proportion at 50%, we need to calculate the margin of error.
The margin of error is determined by dividing the given accuracy within the confidence interval by the square root of the sample size.
The accuracy within which Gallup claims their sample is accurate is within 4 percentage points, 19 in 20 times. This means that the confidence level is 95%.
Using the formula:
Margin of Error = (Z * σ) / √n
where Z is the Z-score for the desired confidence level, σ is the standard deviation, and n is the sample size.
Since the proportion is estimated at 50%, the standard deviation (σ) can be calculated as:
σ = √(p * (1 - p)) = √(0.5 * 0.5) = 0.5
Plugging in the values, we have:
Margin of Error = (Z * 0.5) / √n
To find the closest precision, we need to determine the Z-score for the desired confidence level of 95%.
Using a standard normal distribution table or calculator, the Z-score for a 95% confidence level is approximately 1.96.
Plugging in the values, we have:
0.04 = (1.96 * 0.5) / √n
Rearranging the equation to solve for n:
√n = (1.96 * 0.5) / 0.04
n = [(1.96 * 0.5) / 0.04]^2
n ≈ 600.25
The closest sample size below 600.25 is 600.
Therefore, the closest precision when estimating the population proportion at 50% is approximately 0.025 (b).