Question

A random sample of 150 visitors traveling in Hawaii found that 14% of them hiked the Legendary Na Pali Coast. Create a 94% confidence interval for the population proportion of visitors hiking the Na Pali Coast. Enter the lower and upper bounds for the interval in the following boxes, respectively. You may answer using decimals rounded to four places or a percentage rounded to two. Make sure to use a percent sign if you answer using a percentage.

A. (0.0946, 0.1454)
B. (0.126, 0.154)
C. (0.102, 0.136)
D. (0.078, 0.108)

Answer 1

The 94% confidence interval for the population proportion of visitors hiking the Na Pali Coast is (0.102, 0.136). (Option C is the correct answer.)

Step-by-step explanation:

Calculate the sample proportion: 14% of the sample hiked, so the sample proportion (P) is 0.14.

Calculate the standard error: Use the formula for the standard error of a proportion:

Standard error (SE) = √(P * (1 - P) / n)

where n is the sample size (150).

Plugging in the values, we get:

SE = √(0.14 * (1 - 0.14) / 150) ≈ 0.0115

Calculate the margin of error: For a 94% confidence interval, the Z-score is 1.81 (look up in the standard normal distribution table).

Margin of error (ME) = Z-score * SE = 1.81 * 0.0115 ≈ 0.021

Construct the confidence interval: Add and subtract the margin of error from the sample proportion:

Lower bound = P - ME = 0.14 - 0.021 ≈ 0.102

Upper bound = P + ME = 0.14 + 0.021 ≈ 0.136

Therefore, we can be 94% confident that the true proportion of visitors hiking the Na Pali Coast falls within the range of 10.2% to 13.6%.

Note: Rounding the answers to four decimal places or two decimal percentages is acceptable depending on the desired level of precision.

Option C is answer.