Calculating 95% Confidence Interval for Mean Magnesium Concentration
To calculate a 95% confidence interval for the mean magnesium concentration in this sample, we need to use the following formula for a confidence interval when the population standard deviation is unknown:
\[ \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \( \bar{x} \) is the sample mean,
- \( t_{\frac{\alpha}{2}, n-1} \) is the t-score from the t-distribution for a 95% confidence level with \( n-1 \) degrees of freedom (where \( n \) is the sample size),
- \( s \) is the sample standard deviation, and
- \( n \) is the sample size.
First, we need to calculate the sample mean (\( \bar{x} \)) and the sample standard deviation (s). The data we have is:
\[ 175, 177, 175, 180, 138, 138 \]
Now let's calculate the mean and the standard deviation.
Mean (\( \bar{x} \)):
\[ \bar{x} = \frac{175 + 177 + 175 + 180 + 138 + 138}{6} = \frac{983}{6} \approx 163.83 \]
To calculate the standard deviation, we need to follow these steps:
1. Find the squared difference from the mean for each observation.
2. Sum these squared differences.
3. Divide by the number of observations minus 1 to get the variance.
4. Take the square root of the variance to get the standard deviation.
Let's do these calculations.
1. Calculate the squared differences from the mean:
\[ (175 - 163.83)^2 = 123.69 \]
\[ (177 - 163.83)^2 = 173.69 \]
\[ (175 - 163.83)^2 = 123.69 \]
\[ (180 - 163.83)^2 = 262.89 \]
\[ (138 - 163.83)^2 = 664.89 \]
\[ (138 - 163.83)^2 = 664.89 \]
2. Sum of squared differences:
\[ 123.69 + 173.69 + 123.69 + 262.89 + 664.89 + 664.89 = 2014.74 \]
3. Variance (s^2):
\[ s^2 = \frac{2014.74}{6 - 1} = \frac{2014.74}{5} = 402.948 \]
4. Standard deviation (s):
\[ s = \sqrt{402.948} \approx 20.07 \]
Next, we need to find the t-score for a 95% confidence level with 5 degrees of freedom (since \( n = 6 \)). This value can be found in a t-distribution table or calculated using statistical software. Assuming we are using a t-distribution table or a calculator:
\[ t_{\frac{\alpha}{2}, 5} \text{ at } \alpha = 0.05 \text{ is approximately } 2.571 \]
Finally, calculate the margin of error (ME):
\[ ME = t_{\frac{\alpha}{2}, 5} \left( \frac{s}{\sqrt{n}} \right) = 2.571 \left( \frac{20.07}{\sqrt{6}} \right) \approx 2.571 \left( 8.187 \right) \approx 21.04 \]
Now we can calculate the 95% confidence interval:
\[ \bar{x} \pm ME = 163.83 \pm 21.04 \]
Lower limit:
\[ 163.83 - 21.04 = 142.79 \]
Upper limit:
\[ 163.83 + 21.04 = 184.87 \]
Therefore, the 95% confidence interval for the mean magnesium concentration is approximately (142.79, 184.87) parts per million.
