Question
An environmentalist wants to find out the fraction of oil tankers that have spills each month.
Step 2 of 2 : Suppose a sample of 292
tankers is drawn. Of these ships, 234
did not have spills. Using the data, construct the 80%
confidence interval for the population proportion of oil tankers that have spills each month. Round your answers to three decimal places.
Answers
GPT 3.5
First, calculate the sample proportion of oil tankers that had spills:
Sample proportion (p-hat) = (292 - 234) / 292 = 58 / 292 = 0.199
Next, calculate the standard error:
Standard error = sqrt[(p-hat * (1 - p-hat)) / n] = sqrt[(0.199 * 0.801) / 292] = sqrt(0.159 / 292) = 0.023
Calculate the margin of error using the z-score for 80% confidence (z = 1.282):
Margin of error = z * standard error = 1.282 * 0.023 = 0.029
Finally, construct the confidence interval:
Lower bound = p-hat - margin of error = 0.199 - 0.029 = 0.170
Upper bound = p-hat + margin of error = 0.199 + 0.029 = 0.228
Therefore, the 80% confidence interval for the population proportion of oil tankers that have spills each month is (0.170, 0.228).
Sample proportion (p-hat) = (292 - 234) / 292 = 58 / 292 = 0.199
Next, calculate the standard error:
Standard error = sqrt[(p-hat * (1 - p-hat)) / n] = sqrt[(0.199 * 0.801) / 292] = sqrt(0.159 / 292) = 0.023
Calculate the margin of error using the z-score for 80% confidence (z = 1.282):
Margin of error = z * standard error = 1.282 * 0.023 = 0.029
Finally, construct the confidence interval:
Lower bound = p-hat - margin of error = 0.199 - 0.029 = 0.170
Upper bound = p-hat + margin of error = 0.199 + 0.029 = 0.228
Therefore, the 80% confidence interval for the population proportion of oil tankers that have spills each month is (0.170, 0.228).