To calculate a 95% confidence interval for the mean mass of the dog food, we can follow these steps:
- Calculate the sample mean (\( \bar{x} \)).
- Calculate the sample standard deviation (\( s \)).
- Determine the appropriate t-score for 95% confidence.
- Calculate the margin of error and the confidence interval.
Step 1: Calculate the Sample Mean
The sample mean can be calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
Where \( x_i \) are the sample observations and \( n \) is the sample size.
Given the masses of the containers:
9.8, 10.4, 9.8, 10.0, 10.2, 9.6
The sum of the masses is:
\[ 9.8 + 10.4 + 9.8 + 10.0 + 10.2 + 9.6 = 60.8 \]
Now, calculate the mean:
\[ \bar{x} = \frac{60.8}{6} = 10.1333 \text{ (approximately)} \]
Step 2: Calculate the Sample Standard Deviation
The sample standard deviation \( s \) can be calculated using the formula:
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
First, we calculate each deviation from the mean, square it, and then sum those squared deviations.
- Deviations:
- \( 9.8 - 10.1333 \approx -0.3333 \)
- \( 10.4 - 10.1333 \approx 0.2667 \)
- \( 9.8 - 10.1333 \approx -0.3333 \)
- \( 10.0 - 10.1333 \approx -0.1333 \)
- \( 10.2 - 10.1333 \approx 0.0667 \)
- \( 9.6 - 10.1333 \approx -0.5333 \)
- Squared deviations:
- \( (-0.3333)^2 \approx 0.1111 \)
- \( (0.2667)^2 \approx 0.0711 \)
- \( (-0.3333)^2 \approx 0.1111 \)
- \( (-0.1333)^2 \approx 0.0178 \)
- \( (0.0667)^2 \approx 0.0044 \)
- \( (-0.5333)^2 \approx 0.2844 \)
- Sum of squared deviations:
\[ 0.1111 + 0.0711 + 0.1111 + 0.0178 + 0.0044 + 0.2844 \approx 0.5999 \]
Now, plug into the standard deviation formula:
\[ s = \sqrt{\frac{0.5999}{6 - 1}} = \sqrt{\frac{0.5999}{5}} \approx \sqrt{0.11998} \approx 0.3463 \]
Step 3: Determine the t-score
For a 95% confidence interval with \( n-1 = 5 \) degrees of freedom, we can find the t-score from the t-distribution table. The t-score for 95% confidence and 5 degrees of freedom is approximately:
\[ t_{0.025, 5} \approx 2.571 \]
Step 4: Calculate the Margin of Error and Confidence Interval
The margin of error (ME) can be calculated using the formula:
\[ \text{ME} = t \cdot \frac{s}{\sqrt{n}} \]
\[ \text{ME} = 2.571 \cdot \frac{0.3463}{\sqrt{6}} \approx 2.571 \cdot 0.1414 \approx 0.364 \]
Now, we can calculate the confidence interval:
\[ \text{CI} = \left( \bar{x} - \text{ME}, \bar{x} + \text{ME} \right) \]
\[ \text{CI} = \left( 10.1333 - 0.364, 10.1333 + 0.364 \right) = \left( 9.7693, 10.4973 \right) \]
So, the 95% confidence interval for the mean mass of all such dog food is approximately:
\[ (9.77, 10.50) \text{ (rounded to two decimal places)} \]