A key focus of this week is how to make inferences about populations based on samples. The essential logic lies in comparing a single instance of a statistic such as a sample mean to a distribution of such values. The comparison can lead to one of two conclusions – the sample statistic is either extreme or not extreme. But what are the thresholds for making this kind of judgment call (i.e., whether a value is extreme or not)? This activity explores that question.

The problem is this: You receive a sample containing the ages of 30 students. You are wondering whether this sample is a group of undergraduates (mean age = 20 years) or graduates (mean age = 25 years). To answer this question, you must compare the mean of the sample you receive to a distribution of means from the population. The following fragment of R code begins the solution:

set.seed(2)
sampleSize <- 30
studentPop <- rnorm(20000,mean=20,sd=3)
undergrads <- sample(studentPop,size=sampleSize,replace=TRUE)
grads <- rnorm(sampleSize,mean=25,sd=3)
if (runif(1) >0.5) { testSample <- grads } else { testSample <- undergrads }
mean(testSample)
[1] 25.54158

After you run this code, the variable “testSample” will contain either a sample of undergrads or a sample of grads. The line before last “flips a coin” by generating one value from a uniform distribution (by default the distribution covers 0 to 1) and comparing it to 0.5. The question you must answer with additional code is: Which is it, grad or undergrad? Here are the steps that will help you finish the job:

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