Final answer:
To find the probability that a randomly chosen student will have a GPA over 3.31, we calculate the z-score and use the standard normal distribution table. For a sample of 9 students, we use the sampling distribution of the sample mean and calculate the z-score. The probabilities are approximately 0.6966 and 0.9147, respectively.
Step-by-step explanation:
To find the probability that a randomly chosen student will have a GPA over 3.31, we need to calculate the z-score and use the standard normal distribution table. The formula for calculating the z-score is: z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the population mean, and σ is the population standard deviation.
Using the formula, we have: z = (3.31 - 2.73) / 1.12 = 0.518
Now, we can use the standard normal distribution table or a calculator to find the probability associated with the z-score. In this case, the probability is approximately 0.6966.
To find the probability that the mean GPA of a sample of 9 students will be more than 3.24, we need to use the sampling distribution of the sample mean. Since the population standard deviation is known, we can use the z-score formula. The formula for the standard error of the sample mean is: SE = σ / sqrt(n), where σ is the population standard deviation and n is the sample size.
Using the formula, we have: SE = 1.12 / sqrt(9) = 0.3733
Next, we calculate the z-score using the formula: z = (x - μ) / SE, where x is the value we want to find the probability for and μ is the population mean. In this case, the population mean is the same as before, 2.73. So, z = (3.24 - 2.73) / 0.3733 = 1.37
Now, we can use the standard normal distribution table or a calculator to find the probability associated with the z-score. In this case, the probability is approximately 0.9147.