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Chapter 11: Problem 7

For each of the two samples $$ \begin{array}{l} \text { Sample } 1:\\{8,1,11,0,-2,\\} \\ \text { Sarrpla } 2:\\{-2,0,0,0,2,4,-1\\} \end{array} $$ find a. the sample size, b. the sample mean, c. the sample variance.

Short Answer

Expert verified
Sample 1: Size = 5, Mean = 3.6, Variance = 31.3; Sample 2: Size = 7, Mean 鈮� 0.43, Variance 鈮� 3.95.

Step by step solution

01

Identify the Sample Size for Sample 1

To find the sample size for Sample 1, count the number of elements in the set \(\{8, 1, 11, 0, -2\}\). There are 5 elements in total.
02

Calculate the Sample Mean for Sample 1

The sample mean is calculated by adding all the elements of Sample 1 and dividing by the number of elements. Sum up the numbers: \[ 8 + 1 + 11 + 0 - 2 = 18 \]Now, divide by the sample size, which is 5:\[ \text{Sample Mean} = \frac{18}{5} = 3.6 \]
03

Determine the Sample Variance for Sample 1

First, subtract the sample mean from each element, square the result, and find the average of these squared differences.- Differences: \((8-3.6), (1-3.6), (11-3.6), (0-3.6), (-2-3.6)\) = \(4.4, -2.6, 7.4, -3.6, -5.6\)- Squares: \(19.36, 6.76, 54.76, 12.96, 31.36\)Sum: \(19.36 + 6.76 + 54.76 + 12.96 + 31.36 = 125.2\)Divide by the sample size - 1: \[ \text{Sample Variance} = \frac{125.2}{4} = 31.3 \]
04

Identify the Sample Size for Sample 2

Count the number of elements in Sample 2, which are \(\{-2, 0, 0, 0, 2, 4, -1\}\). There are 7 elements.
05

Calculate the Sample Mean for Sample 2

Add all the elements of Sample 2 and divide by the number of elements. The sum is:\[ -2 + 0 + 0 + 0 + 2 + 4 - 1 = 3 \]Divide by the sample size, which is 7:\[ \text{Sample Mean} = \frac{3}{7} \approx 0.43 \]
06

Determine the Sample Variance for Sample 2

Subtract the sample mean from each element in Sample 2, square the result, and average these squared differences.- Differences: \((-2-0.43), (0-0.43), (0-0.43), (0-0.43), (2-0.43), (4-0.43), (-1-0.43)\) = \(-2.43, -0.43, -0.43, -0.43, 1.57, 3.57, -1.43\)- Squares: \(5.9, 0.18, 0.18, 0.18, 2.46, 12.75, 2.04\)Sum: \(5.9 + 0.18 + 0.18 + 0.18 + 2.46 + 12.75 + 2.04 = 23.7\)Divide by the sample size - 1: \[ \text{Sample Variance} = \frac{23.7}{6} \approx 3.95 \]

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Key Concepts

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

Sample Size
The sample size refers to the number of elements or observations included in a sample. When conducting descriptive statistics, identifying the exact number of items in the sample is essential because it informs all other statistical calculations.
  • For Sample 1, count how many numbers are present: 8, 1, 11, 0, -2. This makes the sample size equal to 5.
  • For Sample 2, the numbers include: -2, 0, 0, 0, 2, 4, -1. Here, the sample size is 7.
In summary, identifying the sample size is straightforward 鈥� simply count the items. This is a crucial starting point for calculating other statistical values, such as the mean and variance.
Sample Mean
The sample mean is a measure of central tendency that tells you the average value of the sample. This is calculated by summing all the data points and then dividing by the total number of points, which is the sample size detected earlier.
To calculate the sample mean for any given data set:
  • For Sample 1, sum up the elements: \(8 + 1 + 11 + 0 - 2 = 18\). Divide by the size of the sample, which is 5: \(\frac{18}{5} = 3.6\). This represents the average value of Sample 1.

  • For Sample 2, the sum is: \(-2 + 0 + 0 + 0 + 2 + 4 - 1 = 3\). Thus, the mean is: \(\frac{3}{7} \approx 0.43\), indicating the typical value in Sample 2 is close to zero.
The sample mean offers a snapshot of the overall trend or central value of the dataset, making it a fundamental concept in descriptive statistics.
Sample Variance
Sample variance is a measure that tells us how spread out or dispersed the values in the sample are. Variance is vital because it gives insights into the consistency of data points relative to the mean.
Here's how to find it:
  • Start by subtracting the sample mean from each data point. Then, square each result.
  • For Sample 1: Differences from the mean \(3.6\) are \(4.4, -2.6, 7.4, -3.6, -5.6\). Squares of these differences are \(19.36, 6.76, 54.76, 12.96, 31.36\).
  • Sum these squares \(19.36 + 6.76 + 54.76 + 12.96 + 31.36 = 125.2\) and divide by one less than the sample size: \(\frac{125.2}{4} = 31.3\).

  • For Sample 2: Subtract the mean \(0.43\) from each element: \(-2.43, -0.43, -0.43, -0.43, 1.57, 3.57, -1.43\). Squares are \(5.9, 0.18, 0.18, 0.18, 2.46, 12.75, 2.04\).
  • The sum is \(23.7\), and dividing this by 6 results in a variance of about \(3.95\).
Understanding sample variance is crucial when assessing the variability within the dataset. A high variance signifies widely varying data points, while a low variance indicates consistency.

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Most popular questions from this chapter

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