Final answer:
a. The sample proportion is 0.16. b. The one-proportion z-interval procedure is appropriate. c. The 99% confidence interval is (0.054, 0.266). d. The margin of error is 0.106.
Step-by-step explanation:
a. Determine the sample proportion:
The sample proportion (p) can be calculated using the formula p = x/n, where x is the number of successes (8 in this case) and n is the sample size (50 in this case).
p = 8/50 = 0.16
b. Decide whether using the one-proportion z-interval procedure is appropriate:
In order to use the one-proportion z-interval procedure, the conditions np > 5 and n(1-p) > 5 must be satisfied, where p is the sample proportion.
Since np = 50 * 0.16 = 8 and n(1-p) = 50 * 0.84 = 42, both conditions are satisfied.
c. Use the one-proportion z-interval procedure to find the confidence interval:
A 99% confidence interval can be calculated using the formula p ± z * sqrt((p * (1-p)) / n), where z is the z-score corresponding to the desired confidence level (2.58 for 99% confidence).
p ± z * sqrt((p * (1-p)) / n)
0.16 ± 2.58 * sqrt((0.16 * 0.84) / 50)
0.16 ± 0.106
Confidence interval: (0.054, 0.266)
d. Find the margin of error and express the confidence interval:
The margin of error can be calculated by subtracting the lower limit of the confidence interval from the sample proportion.
Margin of error = 0.16 - 0.054 = 0.106
The confidence interval can be expressed as p ± margin of error.
Confidence interval: (0.054, 0.266)