Question

Let X denote the amount of time it takes for a student to complete a quiz a ECU11022. X is normally distributed with a standard deviation of 12 minutes, i.e., X∼N(μ,90). The lecturer assumes that the mean for the entire class is 45 minutes, i.e., μ=45. You would like to investigate whether the assumption is true. Based on what you have learned in class, you take a random sample of 30 students, ask them the time it

a. Provide an expression for the distribution of the sample mean.

Answer 1

The expression for the distribution of the sample mean is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

In this question, we are investigating whether the assumption that the mean time for the entire class to complete a quiz is 45 minutes is true. To do this, we take a random sample of 30 students and ask them the time it took to complete the quiz. The distribution of the sample mean can be described by the Central Limit Theorem, which states that the distribution of sample means is approximately normal, regardless of the shape of the population distribution. The mean of the sample mean distribution is equal to the population mean (45 minutes) and the standard deviation of the sample mean distribution is equal to the population standard deviation divided by the square root of the sample size (12 minutes / √30).