Final answer:
To solve this problem, we use the normal distribution since the sample size is large. For part (a), the probability that the sample proportion is between 0.163 and 0.172 is approximately 0.0918. For part (b), the probability that the sample proportion is greater than 0.055 is approximately 0.9783.
Step-by-step explanation:
To solve this problem, we will use the normal distribution since the sample size is large.
(a) To find the probability that the sample proportion is between 0.163 and 0.172, we will calculate the z-scores for both values:
Z1 = (0.163 - 0.1) / sqrt((0.1)(0.9)/100) = 1.18,
Z2 = (0.172 - 0.1) / sqrt((0.1)(0.9)/100) = 1.3.
Using a standard normal distribution table or calculator, we find the area under the curve between Z = 1.18 and Z = 1.3 is approximately 0.0918.
(b) To find the probability that the sample proportion is greater than 0.055, we will calculate the z-score for this value:
Z = (0.055 - 0.1) / sqrt((0.1)(0.9)/100) = -2.02.
Using a standard normal distribution table or calculator, we find the area under the curve to the right of Z = -2.02 is approximately 0.9783.