Final answer:
To find the probability of a sample mean between 12.2 and 12.8 from a population with a discrete uniform distribution, we can use the Central Limit Theorem and estimate the population standard deviation. We then calculate the standard error and use z-scores to find the probability from a standard normal distribution table or calculator.
Step-by-step explanation:
To find the probability that a random sample of size 96, selected with replacement, will yield a sample mean greater than 12.2 but less than 12.8, we can use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough. Since the population distribution is discrete uniform, we will treat it as an approximation to a continuous distribution.
To calculate the probability, we first need to calculate the standard deviation of the sample mean. The formula for the standard deviation of the sample mean, referred to as the standard error, is given by:
Standard Error = Population Standard Deviation / Square Root of Sample Size
In this case, the population standard deviation is not given, so we cannot calculate the exact standard error. However, we can use the fact that the population mean is 13 to estimate the population standard deviation as the distance between the mean and the endpoints of the distribution, which in this case is 13 - 12 = 1.
Using this estimated population standard deviation, we can calculate the standard error and then use it to find the z-scores corresponding to the sample mean values of 12.2 and 12.8. From the z-scores, we can find the probability using a standard normal distribution table or calculator.