Statistical Inference – Quiz 2

Quiz Question Explanations

Question 1

The distribution of sample means was constructed as follows: First 25 students were randomly sampled from the population. The average score of those 25 students was calculated and saved. Then I repeated this sampling-averaging process 2,000 times. The blue graph shows the histogram of sample averages. In the real world, you will typically only take a single sample. But by virtue of the central limit theorem you can predict the shape of the sample means distribution, and make statements about the likelihood that your sample estimate lies a certain distance from the true population mean. Of course, you won’t know where in the distribution of sample means your estimate actually lies.

Question 2

If the sample size increased from 25 to 50 students, then the likelihood that the sample mean is far from the population mean falls. In other words, the standard error of the sample mean distribution falls. This implies that the standard error bars and range of sample means becomes more tightly clustered around the population mean.

The population mean and the standard deviation of test scores are characteristics of the population, and thus are unaffected by the sample mean estimates.

Question 3

If all students in the population were sampled, then the population mean could be exactly calculated every time. In other words, the sample mean would always equal the population mean. The standard error would be zero in this case. (The standard deviation, which is a characteristic of the population data, would remain unchanged).