Question

Consider a population consisting of the following five values, which represent the number of video rentals during the academic year for each of five housemates: \(\begin{array}{lll} 8 & 14&16 & 10 & 11\end{array}\) a. Compute the mean of this population. b. Select a random sample of size 2 by writing the numbers on slips of paper, mixing them, and then selecting \(2 .\) Compute the mean of your sample. c. Repeatedly select samples of size 2 , and compute the \(\bar{x}\) value for each sample until you have the results of 25 samples. d. Construct a density histogram using the \(25 \bar{x}\) values. Are most of the \(\bar{x}\) values near the population mean? Do the \(\bar{x}\) values differ a lot from sample to sample, or do they tend to be similar?

Short answer

Due to the exercise set-up, no concrete numeric answer can be given. The result of this exercise is more about understanding the concept of population mean, sample mean, and how to interpret those values.

Step-by-step solution

Compute the population mean

The population consists of the number of video rentals: 8, 14, 16, 10, and 11. We calculate the mean by adding each number and then dividing by the total count. So, \(\(Mean = \frac{8 + 14 + 16 + 10 + 11}{5}\)

Compute the mean of a sample of size 2

You need to randomly select two numbers from the population and then calculate their mean. This process can give a different result each time it's executed depending on the numbers chosen.

Compute the means of 25 samples

You need to repeat the process in step 2, 25 times and calculate the mean for each sample.

Construct a density histogram

The density histogram can be constructed using the \(25 \bar{x}\) values. The histogram will provide a visual representation of the means from the 25 samples.

Analyze the results

The goal is to explore whether most sample means are close to the population mean, and whether there are significant differences in the sample means. This can be interpreted by looking at the histogram. If most of the sample means are close to the population mean, this suggests that the sample mean is a good estimator of the population mean. If the bar heights for different sample means don't vary much, it could suggest that sample means tend not to differ a lot.