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)

Answer 1

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.