91Ó°ÊÓ

Consider a population consisting of the following five values, which represent the number of DVD rentals during the academic year for each of five housemates: $$ \begin{array}{lllll} 8 & 14 & 16 & 10 & 11 \end{array} $$ a. Compute the mean of this population. b. Select a random sample of size 2 by writing the five numbers in this population on slips of paper, mixing them, and then selecting two. 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 \(\bar{x}\) values for 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?

Step by step solution

01

Calculate Population Mean

To calculate the mean, sum up all the values and then divide by the number of values. Given values are 8, 14, 16, 10, and 11. Adding these gives a total of \(8 + 14 + 16 + 10 + 11 = 59\). As there are five values in our dataset, the mean is \(59 / 5 = 11.8\).

02

Choose Random Sample & Calculate Sample Mean

Assuming the two numbers randomly selected are 14, 16. Their mean is calculated as \( (14 + 16) / 2 = 15 \). This process will vary depending on the values randomly chosen each time.

03

Repeatedly Select Samples & Compute Mean

This step will demonstrate the process using just one sample due to space constraints. However, this process should be repeated 25 times. For instance, assume the next sample is 10 and 11, the mean is \( (10 + 11) / 2 = 10.5 \).

04

Construct a Density Histogram & Analyze

After repeating the process 25 times, the sample means should be plotted on a histogram. Count the number of occurrences of each mean to construct the bars of the histogram. This will serve to visualize the distribution of the means. By comparing these sample means to population mean, it can be observed whether most of sample means are near the population mean or not, and whether the sample means vary a lot or are relatively stable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean

The population mean is an important statistical measure that provides an average of all the values present in a dataset. In this example, we are working with a dataset of DVD rentals: 8, 14, 16, 10, and 11. To find the population mean:

• First, add together all the values in the dataset: \(8 + 14 + 16 + 10 + 11 = 59\).
• Next, divide the total by the number of values, which is 5 in this case.
• The mathematical expression for this is \(\text{Population Mean} = \frac{59}{5} = 11.8\).

The population mean of 11.8 gives us a central point which represents the average number of DVD rentals by the housemates.

Sample Mean

The sample mean is an estimate of the population mean, derived from a smaller subgroup or portion of the data called a sample. For example, if we randomly selected 2 housemates and calculated their average DVD rentals, we would compute the sample mean.

• Let's assume that our first random sample consists of values 14 and 16.
• To find the sample mean, add 14 and 16, which equals 30, then divide by 2.
• Mathematically, \(\text{Sample Mean} = \frac{14 + 16}{2} = 15\).

Since randomness can create variation, repeating the sampling process with different groups will often yield different sample means. This demonstrates how each sample mean is an estimate rather than a fixed value.

Density Histogram

Constructing a density histogram allows us to visually represent the distribution of sample means. It shows us how often each sample mean value appears across all samples:

• After collecting 25 different sample means, a density histogram can be plotted.
• The horizontal axis represents the sample mean values.
• The vertical axis, or y-axis, represents the frequency or density of those values.
• The bars are drawn to show how many times each mean occurs.

By analyzing the density histogram, we can observe patterns such as:

• Are most sample means close to the population mean of 11.8?
• Do they show a lot of variation or are they consistent?

This visualization technique helps us understand the stability and distribution of sample means relative to the population mean.