Final answer:
Statistics within Mathematics, covering topics like sample distributions, variance, and the effects of sample size on data analysis, utilizing random sampling and the Central Limit Theorem principles.
Step-by-step explanation:
The student's question is related to the subject of Statistics, which is a branch of Mathematics. Specifically, the question addresses concepts such as sampling distributions, sample size (n), population mean (μ), and population standard deviation (σ), as well as the calculation of sample means and the construction of histograms.
When discussing why we divide by (n - 1) instead of n when calculating the sample variance, it's important to mention that this is done to obtain an unbiased estimate of the population variance. Dividing by (n - 1) corrects for the bias that occurs when using sample data to estimate a population parameter, a concept known as Bessel's correction.
Regarding the collection and analysis of data, the use of a random number generator for selecting samples is a random sampling technique. After collecting the data, calculating the sample means, and sketching histograms, we can discuss the effects of sample size on the shape of the distribution.
As the sample size increases, the Central Limit Theorem predicts that the distribution of the sample means will approach a normal distribution, even if the original population distribution is not normal. This results in smaller standard deviation and narrower histograms as n increases.
Finally, when comparing sample means for different sizes (such as n = 5 and n = 10), we expect them to be close to the theoretical mean, μ, because the expected value of the sample mean is the population mean. The distribution of sample means with a larger n will have a smaller standard deviation due to the decreased variability in larger samples.