Question

Suppose two samples of values are taken from a population $$ a: 8,2,4,7,5 \text { and } b: 6,9,10,5,7,8,2,5 $$ (a) Find \(\bar{a}\) and \(\bar{b}\). (b) Find the mean of the sample you get by combining the two samples. (c) Is the mean of the combined sample equal to the mean of the two values \(\bar{a}\) and \(\bar{b}\) ? (d) Explain why the means in (c) are the same or different.

Short answer

Short Answer: The means are different because when calculating the combined mean, we give equal weight to each element in the combined sample, while when calculating the mean of two initial samples' means, we give equal weight to each sample's mean, regardless of the sample size. The larger sample influences the combined mean more than the smaller sample, resulting in a differing combined mean compared to the mean of the two initial samples' means.

Step-by-step solution

Find the mean of sample \(a\) and sample \(b\)

To find the mean of a sample, we add all the values in the sample and divide by the number of values in that sample. So, for sample \(a\) and sample \(b\), we will calculate the mean as follows: $$ \bar{a} = \frac{8+2+4+7+5}{5} = \frac{26}{5} = 5.2 \\ \bar{b} = \frac{6+9+10+5+7+8+2+5}{8} = \frac{52}{8} = 6.5 $$

Find the mean of the combined sample

To find the mean of the combined sample, we first combine the two samples: $$ a \cup b = 8,2,4,7,5,6,9,10,5,7,8,2,5 $$ Next, we find the mean of this combined sample: $$ \bar{c} = \frac{8+2+4+7+5+6+9+10+5+7+8+2+5}{13} = \frac{78}{13} = 6 $$

Comparing the combined mean with the means of the initial samples

Now, we need to check if the mean of the combined sample is equal to the mean of the two initial samples' means. We have found these means in Step 1 and Step 2: $$ \bar{a} = 5.2 \\ \bar{b} = 6.5 \\ \bar{c} = 6 $$ To check if the combined mean is equal to the mean of the two initial samples, we can calculate this mean: $$ \frac{\bar{a}+\bar{b}}{2} = \frac{5.2+6.5}{2} = \frac{11.7}{2} = 5.85 $$

Explain why the means in (c) are the same or different

From step 3, we see that the mean of the combined sample, \(\bar{c}=6\), is not equal to the mean of the two initial samples' means, which is 5.85. The reason for this difference is that the sample sizes of a and b are different, which affects the combined mean. When calculating the combined mean, we give equal weight to each element in the combined sample, while when calculating the mean of two initial samples' means, we give equal weight to each sample's mean, regardless of the sample size. So, the larger sample influences the combined mean more than the smaller sample, which results in a different mean for the combined sample compared to the mean of two initial samples' means.

Key concepts

Population Sampling

Sampling from a population is a fundamental process in statistics. It involves selecting a group of individuals, items, or data points from a larger set called the population. This helps researchers and analysts make inferences about the entire population without needing to study every single member.
There are several types of sampling methods, including:

• Simple Random Sampling: Every individual has an equal chance of being selected.
• Systematic Sampling: Every nth item in a list is chosen.
• Stratified Sampling: The population is divided into strata, and random samples are taken from each stratum.
• Cluster Sampling: The entire population is divided into clusters, and a whole cluster is randomly selected.

Each sampling method has its own applications and is chosen based on research needs and the nature of the population. The main goal is to obtain a sample that is representative of the population, allowing for accurate predictions.

Mean Calculation

The mean, often called the average, is a measure of central tendency that provides a single value that summarizes a set of numbers. To calculate the mean, you sum up all the numerical values in a data set and divide by the total number of values.
For example, given data sets:

• Sample a: 8, 2, 4, 7, 5
• Sample b: 6, 9, 10, 5, 7, 8, 2, 5

To determine the mean of sample "a", sum the values (8 + 2 + 4 + 7 + 5 = 26) and then divide by the number of values (5), resulting in \(\bar{a} = \frac{26}{5} = 5.2\). Similarly, calculate for sample "b": \(\bar{b} = \frac{52}{8} = 6.5\).
Mean calculation is essential for understanding data distribution and is widely used for statistical analyses, making it a crucial component of data interpretation.

Weighted Average

A weighted average takes into account the varying degrees of importance of the numbers in a data set. This is different from a regular mean where each value is treated equally.
In the context of combining sample means, the sample sizes act like weights. For samples with different sizes, it is often more appropriate to use a weighted mean to accurately represent the combined data.
To find the weighted average of sample means, multiply each mean by its sample size (weight) and then divide the sum by the total of the weights:

• Sample "a" has a mean of 5.2 with a sample size of 5.
• Sample "b" has a mean of 6.5 with a sample size of 8.

Calculating the weighted mean: \ \( \bar{w} = \frac{(5.2 \times 5) + (6.5 \times 8)}{5 + 8} = \frac{26 + 52}{13} = 6 \)\
The weighted average is helpful when combining different groups as it takes into account both the mean and size of each group, providing a more balanced overall average.