Math Problem Statement
Solution
To solve the given tasks, I'll first extract the relevant data from the table and proceed with the required calculations.
Step 1: Extract Data
- Volume of water delivered (mL): 50 mL for all trials.
- Mass of beaker + water delivered (g):
- Trial 1: 96.190 g
- Trial 2: 93.964 g
- Trial 3: 92.850 g
- Mass of water delivered (g):
- Trial 1: 48.445 g
- Trial 2: 46.219 g
- Trial 3: 45.105 g
- Density of water (g/mL):
- Trial 1: 0.9689 g/mL
- Trial 2: 0.9244 g/mL
- Trial 3: 0.9021 g/mL
Step 2: Calculating the Average Density
To find the average density:
Step 3: Calculating the Standard Deviation
The standard deviation (σ) of the densities is calculated as: Where are the individual densities, is the mean density, and is the number of trials.
Plugging in the values:
Step 4: Calculating the 90% Confidence Limit for the Average Density
The 90% confidence interval can be calculated using: For a 90% confidence level and , the t-value (from the t-distribution table) is approximately 1.886.
Thus, the confidence limit is: So, the 90% confidence interval is approximately to g/mL.
Step 5: Calculating the Relative Error
The relative error can be calculated as:
Summary
- Average Density: 0.9318 g/mL
- Standard Deviation: 0.0339 g/mL
- 90% Confidence Interval: 0.8948 to 0.9688 g/mL
- Relative Error: 3.64%
Would you like further details or explanations on any of these steps?
Related Questions:
- How does increasing the number of trials affect the accuracy of the average density?
- What is the significance of using a t-value in calculating the confidence interval?
- How would the results change if the volume of water delivered was different for each trial?
- Why is it important to use the standard deviation when analyzing experimental data?
- How can outliers affect the calculation of standard deviation and mean in this experiment?
Tip:
When calculating confidence intervals, always ensure you have the correct t-value based on your sample size and desired confidence level.
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Math Problem Analysis
Mathematical Concepts
Statistics
Measurement
Standard Deviation
Confidence Interval
Relative Error
Formulas
Average density formula: (sum of densities) / (number of trials)
Standard deviation formula: sqrt((1/(N-1)) * sum((xi - x̄)^2))
Confidence limit formula: x̄ ± t(σ/√N)
Relative error formula: (Standard deviation / Average density) * 100
Theorems
t-distribution for confidence intervals
Error propagation in measurements
Suitable Grade Level
Grades 11-12 (Advanced High School or Introductory College Level)
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