Out of 167

Question

randomly selected adults in the United States who were surveyed, 88
exercise on a regular basis. Construct a 90%
confidence interval for the proportion of all adults in the United States who exercise on a regular basis. Round to three decimal places.

Answer 1

To construct a 90% confidence interval, we can use the formula for calculating the confidence interval for a proportion:

CI = p-hat ± z * sqrt[(p-hat * (1 - p-hat)) / n]

Where:
CI = Confidence Interval
p-hat = Sample proportion (88/167 = 0.5269)
z = Z-value for 90% confidence level (z = 1.645 for 90% confidence level)
n = Sample size (167)

Plugging in the values:

CI = 0.5269 ± 1.645 * sqrt[(0.5269 * (1 - 0.5269)) / 167]
CI = 0.5269 ± 1.645 * sqrt[(0.5269 * 0.4731) / 167]
CI = 0.5269 ± 1.645 * sqrt[0.2497 / 167]
CI = 0.5269 ± 1.645 * sqrt[0.001495]
CI = 0.5269 ± 1.645 * 0.03866
CI = 0.5269 ± 0.0636

Therefore, the 90% confidence interval for the proportion of all adults in the United States who exercise on a regular basis would be (0.463 to 0.591), rounded to three decimal places.