In a cafe, the owner surveyed 200 customers about their favorite drink; 42 customers said that they prefer caramel macchiato. Estimate the population proportion using two standard errors. Round answers to two decimal places.(1 point)
To estimate the population proportion using two standard errors, we first find the sample proportion of customers who prefer caramel macchiato.
Sample proportion = Number of customers who prefer caramel macchiato / Total number of customers surveyed
Sample proportion = 42 / 200
Sample proportion = 0.21
Next, we calculate the standard error:
Standard error = sqrt[(sample proportion * (1 - sample proportion)) / sample size]
Standard error = sqrt[(0.21 * 0.79) / 200]
Standard error = sqrt(0.1659 / 200)
Standard error = sqrt(0.0008295)
Standard error = 0.02879
To estimate the population proportion using two standard errors, we calculate the margin of error:
Margin of error = Standard error * 2
Margin of error = 0.02879 * 2
Margin of error = 0.05758
Finally, we calculate the lower and upper bounds of the confidence interval:
Lower bound = Sample proportion - Margin of error
Lower bound = 0.21 - 0.05758
Lower bound = 0.15242
Upper bound = Sample proportion + Margin of error
Upper bound = 0.21 + 0.05758
Upper bound = 0.26758
Therefore, the estimated population proportion of customers who prefer caramel macchiato using two standard errors is between 0.15 and 0.27.