I would really appreciate help with this.
In a survey of 1000 large corporations, 252 said that, given a choice between a job candidate who smokes and an equally qualified nonsmoker, the nonsmoker would get the job.
Find a 0.95 confidence interval for p. (Round your answers to three decimal places.)
Lower Limit
Upper Limit
What is the margin of error based on a 95% confidence interval? (Round your answer to three decimal places.)
To find a 0.95 confidence interval for the proportion, p, we can use the formula:
Lower Limit = p - Margin of Error
Upper Limit = p + Margin of Error
First, we need to calculate the point estimate, which is the proportion of corporations that said a nonsmoker would get the job. In this case, the point estimate, p̂, is:
p̂ = (number of corporations saying nonsmoker would get the job) / (total number of corporations surveyed)
= 252 / 1000
= 0.252
Next, we need to calculate the standard error, which measures the average deviation of the sample proportion from the true population proportion. The formula for the standard error, SE, is:
SE = sqrt((p̂ * (1 - p̂)) / n)
Where:
- p̂ is the point estimate,
- (1 - p̂) is the complement of the point estimate,
- n is the sample size.
Using the given values:
- p̂ = 0.252
- n = 1000
SE = sqrt((0.252 * (1 - 0.252)) / 1000)
= sqrt(0.188496 / 1000)
= sqrt(0.000188496)
≈ 0.013735
To find the margin of error, we need to multiply the standard error by the critical value from the standard normal distribution for a 95% confidence level. The critical value is approximately 1.96 for a 95% confidence level.
Margin of Error = Critical value * Standard Error
= 1.96 * 0.013735
≈ 0.0269
Finally, we can calculate the lower and upper limits of the confidence interval:
Lower Limit = p̂ - Margin of Error
= 0.252 - 0.0269
≈ 0.225
Upper Limit = p̂ + Margin of Error
= 0.252 + 0.0269
≈ 0.279
Therefore, the 0.95 confidence interval for p is approximately:
Lower Limit: 0.225
Upper Limit: 0.279
To find the margin of error based on a 95% confidence interval, we have already calculated it as 0.0269.