Using the results of a Newsweek poll conducted in March of 2009, a 90% confidence interval on the proportion of adult Americans who have a favorable opinion of Barack Obama is given by (0.699, 0.741). What is the margin of error?
To calculate the margin of error, you first need to determine the sample proportion and the sample size. Given that the confidence interval is for the proportion of adult Americans with a favorable opinion of Barack Obama, we can use the midpoint of the confidence interval as the sample proportion.
The midpoint of the confidence interval is the average of the upper and lower bounds:
Midpoint = (0.699 + 0.741) / 2 = 0.72
The margin of error can be calculated using the formula:
Margin of Error = Critical Value * Standard Error
To find the critical value, we need to determine the z-score for a 90% confidence level. A 90% confidence level corresponds to a 5% significance level, meaning that there is a 5% chance of the sample proportion being outside the confidence interval.
To determine the critical value, we can use a standard normal distribution table or a statistical calculator. A 90% confidence level corresponds to a z-score of approximately 1.645.
The standard error can be calculated using the formula:
Standard Error = sqrt((p_hat * (1 - p_hat)) / n)
Using the midpoint as the sample proportion and the given confidence interval, we can use the midpoint formula to estimate p_hat:
p_hat = 0.72
The sample size (n) is not given in the question, so we cannot calculate the exact margin of error without that information. The margin of error is generally inversely proportional to the square root of the sample size. The larger the sample size, the smaller the margin of error.
Therefore, without the value of the sample size, we cannot calculate the margin of error precisely.