https://file.aiquickdraw.com/mathbot/file/1728974844642m0v7agmz.jpg

The table you've provided includes information for calculating the average density, standard deviation, 90% confidence limit, and relative error based on three trials. Let's break down the required calculations.

### Data from the Table:
- **Trial 1**: 
  - Volume of water: 12.00 mL
  - Mass of flask + water: 104.772 g
  - Mass of water delivered: 10.719 g
  - Density: 0.893 g/mL

- **Trial 2**: 
  - Volume of water: 12.00 mL
  - Mass of flask + water: 116.886 g
  - Mass of water delivered: 22.833 g
  - Density: 1.903 g/mL

- **Trial 3**: 
  - Volume of water: 12.00 mL
  - Mass of flask + water: 129.658 g
  - Mass of water delivered: 35.605 g
  - Density: 2.967 g/mL

### 1. **Average Density (g/mL):**
To calculate the average density, sum the densities from each trial and divide by the number of trials.
\[
\text{Average Density} = \frac{(0.893 + 1.903 + 2.967)}{3}
\]
\[
\text{Average Density} = \frac{5.763}{3} = 1.921 \text{ g/mL}
\]

### 2. **Standard Deviation of Density:**
The standard deviation (\( \sigma \)) is calculated using:
\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n - 1}}
\]
Where \( x_i \) are the individual densities, \( \mu \) is the mean, and \( n \) is the number of trials.

First, calculate the squared differences:
- \( (0.893 - 1.921)^2 = 1.058^2 = 1.119364 \)
- \( (1.903 - 1.921)^2 = 0.018^2 = 0.000324 \)
- \( (2.967 - 1.921)^2 = 1.046^2 = 1.094116 \)

Now sum the squared differences:
\[
\sum = 1.119364 + 0.000324 + 1.094116 = 2.213804
\]
Finally, divide by \( n-1 \) and take the square root:
\[
\sigma = \sqrt{\frac{2.213804}{2}} = \sqrt{1.106902} = 1.052 \text{ g/mL}
\]

### 3. **90% Confidence Limit for the Average Density:**
To calculate the 90% confidence interval, use:
\[
\text{Confidence Interval} = \mu \pm t_{\alpha/2}

Calculate the average density, standard deviation, 90% confidence limit, and relative error based on the provided table with water delivery trials.

Learn how to calculate the average density, standard deviation, 90% confidence interval, and relative error using experimental data from multiple trials. This tutorial includes formulas and step-by-step solutions for students working on lab experiments.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation