U D 19. Using the computing formula calculate the sample variance and sample standard deviation for the following Forecast data. Also, calculate the mean. 6-day Forecast: 93, 100, 109, 113, 111, 104 ||EX= n = X= X X² Place your answers in the boxes, use the symbols. Sample Variance Sample Standard Deviation
U D 19. Using the computing formula calculate the sample variance and sample standard deviation for the following Forecast data. Also, calculate the mean. 6-day Forecast: 93, 100, 109, 113, 111, 104 ||EX= n = X= X X² Place your answers in the boxes, use the symbols. Sample Variance Sample Standard Deviation
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Chapter 4**
**19. Using the computing formula, calculate the sample variance and sample standard deviation for the following Forecast data. Also, calculate the mean.**
**6-day Forecast: 93, 100, 109, 113, 111, 104**
| X | X² |
|----|-------|
| | |
| | |
| | |
| | |
| | |
| | |
| ΣX=| |
| n=| |
|X̄ =| |
**Place your answers in the boxes, use the symbols.**
| Sample Variance | Sample Standard Deviation |
|-------------------------|-----------------------------|
| | |
**Homework Packet One - Riess**
---
**Instructions for completion:**
1. **List each value (X) in the 6-day forecast data** in the left column of the table.
2. **Square each value (X²)** and list the squared values in the right column of the table.
3. **Compute the sum of the values (ΣX)** and write it at the bottom of the left column.
4. **Compute the sum of the squared values (ΣX²)** and write it at the bottom of the right column.
5. **Determine the sample size (n)** and record it.
6. **Calculate the mean (X̄)** using the formula X̄ = ΣX / n and write it in the space provided.
7. **Calculate the sample variance** using the formula for variance:
\[
s^2 = \frac{Σ(X - X̄)²}{n-1}
\]
and place the result in the Sample Variance box.
8. **Calculate the sample standard deviation** by taking the square root of the sample variance:
\[
s = \sqrt{s^2}
\]
and place the result in the Sample Standard Deviation box.
This exercise aims to help you understand and apply statistical methods to forecast data analysis, enhancing your skills in data interpretation and variability measurement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55e5aa3a-0d07-45ba-a16f-5f15ce7e32c0%2F28053c5b-1eb1-4154-8648-9bc46f595a20%2Frrbr6nb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chapter 4**
**19. Using the computing formula, calculate the sample variance and sample standard deviation for the following Forecast data. Also, calculate the mean.**
**6-day Forecast: 93, 100, 109, 113, 111, 104**
| X | X² |
|----|-------|
| | |
| | |
| | |
| | |
| | |
| | |
| ΣX=| |
| n=| |
|X̄ =| |
**Place your answers in the boxes, use the symbols.**
| Sample Variance | Sample Standard Deviation |
|-------------------------|-----------------------------|
| | |
**Homework Packet One - Riess**
---
**Instructions for completion:**
1. **List each value (X) in the 6-day forecast data** in the left column of the table.
2. **Square each value (X²)** and list the squared values in the right column of the table.
3. **Compute the sum of the values (ΣX)** and write it at the bottom of the left column.
4. **Compute the sum of the squared values (ΣX²)** and write it at the bottom of the right column.
5. **Determine the sample size (n)** and record it.
6. **Calculate the mean (X̄)** using the formula X̄ = ΣX / n and write it in the space provided.
7. **Calculate the sample variance** using the formula for variance:
\[
s^2 = \frac{Σ(X - X̄)²}{n-1}
\]
and place the result in the Sample Variance box.
8. **Calculate the sample standard deviation** by taking the square root of the sample variance:
\[
s = \sqrt{s^2}
\]
and place the result in the Sample Standard Deviation box.
This exercise aims to help you understand and apply statistical methods to forecast data analysis, enhancing your skills in data interpretation and variability measurement.
![### Chapter 4 - Statistical Concepts
#### 20. **Bias in Sample Variance and Sample Standard Deviation**
Without some adjustment, sample variance and sample standard deviation are "biased." Define the term biased, in relation to samples, and explain how this bias is corrected in the formula for the unbiased estimated population variance and/or standard deviation.
**Answer:**
- **Bias**: In the context of statistics, a biased estimator is one that does not estimate the parameter accurately. For instance, the sample variance tends to underestimate the population variance because it uses the sample mean rather than the population mean.
- **Correction**: To correct this bias, the formula for the sample variance divides by \(n-1\) instead of \(n\). This adjustment is known as Bessel's correction. Mathematically, the unbiased estimator for variance is:
\[
\text{Unbiased Variance} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2
\]
#### 21. **Estimated Standard Deviation Calculation**
Calculate the **estimated standard deviation** for the following sets of sums and squared sums.
| $\sum{X}$ | $\sum{X^2}$ | $n$ |
|-----------|--------------|-----|
| 262.0 | 5227.0 | 25 |
| 96.5 | 978.0 | 10 |
| 487.0 | 8201.0 | 33 |
**Steps for Calculation:**
1. **Calculate Mean (\(\bar{X}\))**:
\[
\bar{X} = \frac{\sum{X}}{n}
\]
2. **Calculate the Variance**:
\[
\text{Variance} = \frac{\sum{X^2} - (\sum{X})^2 / n}{n - 1}
\]
3. **Calculate Standard Deviation**:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]
**Examples:**
1. For \( \sum{X} = 262.0, \sum{X^2} = 5227.0, n = 25 \):
- \( \bar{X} = \frac{262.0}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55e5aa3a-0d07-45ba-a16f-5f15ce7e32c0%2F28053c5b-1eb1-4154-8648-9bc46f595a20%2Fgquwt1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Chapter 4 - Statistical Concepts
#### 20. **Bias in Sample Variance and Sample Standard Deviation**
Without some adjustment, sample variance and sample standard deviation are "biased." Define the term biased, in relation to samples, and explain how this bias is corrected in the formula for the unbiased estimated population variance and/or standard deviation.
**Answer:**
- **Bias**: In the context of statistics, a biased estimator is one that does not estimate the parameter accurately. For instance, the sample variance tends to underestimate the population variance because it uses the sample mean rather than the population mean.
- **Correction**: To correct this bias, the formula for the sample variance divides by \(n-1\) instead of \(n\). This adjustment is known as Bessel's correction. Mathematically, the unbiased estimator for variance is:
\[
\text{Unbiased Variance} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2
\]
#### 21. **Estimated Standard Deviation Calculation**
Calculate the **estimated standard deviation** for the following sets of sums and squared sums.
| $\sum{X}$ | $\sum{X^2}$ | $n$ |
|-----------|--------------|-----|
| 262.0 | 5227.0 | 25 |
| 96.5 | 978.0 | 10 |
| 487.0 | 8201.0 | 33 |
**Steps for Calculation:**
1. **Calculate Mean (\(\bar{X}\))**:
\[
\bar{X} = \frac{\sum{X}}{n}
\]
2. **Calculate the Variance**:
\[
\text{Variance} = \frac{\sum{X^2} - (\sum{X})^2 / n}{n - 1}
\]
3. **Calculate Standard Deviation**:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]
**Examples:**
1. For \( \sum{X} = 262.0, \sum{X^2} = 5227.0, n = 25 \):
- \( \bar{X} = \frac{262.0}{
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