Final answer:
The sampling distribution for the proportion of orders shipped within three working days will be approximately normal due to the Central Limit Theorem. The mean of the sampling distribution is 0.92, and the standard deviation can be calculated using π(1-π)/n. To find the probability of at least 95% being shipped on time, use a normal distribution with the found mean and standard deviation.
Step-by-step explanation:
When you select a simple random sample (SRS) of 200 orders from the past week to audit the proportion that were shipped within three working days, you are dealing with the concept of a sampling distribution. The sampling distribution of the sample proportion (πhat) will tend to follow the following characteristics based on the given information:
a. Shape: The shape of the sampling distribution will be approximately normal. This can be determined using the Central Limit Theorem since nπ(1-π) is greater than 10, where n is the sample size (200) and π is the population proportion (0.92).
b. Mean: The mean (μπhat) of the sampling distribution for πhat is equal to the population proportion, which in this case is 0.92.
c. Standard Deviation: The standard deviation (σπhat) of the sampling distribution for πhat is √[π(1-π)/n], which equals √[0.92(0.08)/200].
d. Probability: To find the probability that 95% or more of the orders are delivered within three working days, you would use the z-score formula (X - μπhat) / σπhat. Calculate the z-score corresponding to 0.95, and then use standard normal tables or technology to find the probability.