Answer :
To address this question, we need to calculate a confidence interval for the mean concentration of fine particulates in the air. Here's a detailed, step-by-step solution:
1. Determine the necessary statistical parameters:
- Sample mean (\(\bar{x}\)): 12 \(\mu g / m^3\)
- Sample size (n): 196
- Sample standard deviation (s): 3.5 \(\mu g / m^3\)
- Confidence level: 99.7%
2. Convert the confidence level to a z-score:
- For a 99.7% confidence level, traditionally we use a z-score associated with this confidence level. In this case, the z-score is approximately 3.
3. Calculate the standard error of the mean (SE):
- The standard error is given by the formula:
[tex]\[ SE = \frac{s}{\sqrt{n}} \][/tex]
- Plugging in the values:
[tex]\[ SE = \frac{3.5}{\sqrt{196}} = \frac{3.5}{14} = 0.25 \][/tex]
4. Calculate the margin of error (ME):
- The margin of error is given by the formula:
[tex]\[ ME = z \times SE \][/tex]
- Plugging in the values:
[tex]\[ ME = 3 \times 0.25 = 0.75 \][/tex]
5. Determine the confidence interval:
- The confidence interval can be found using the formula:
[tex]\[ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) \][/tex]
- Plugging in the values:
[tex]\[ \text{Confidence Interval} = (12 - 0.75, 12 + 0.75) = (11.25, 12.75) \][/tex]
Therefore, the actual concentration of fine particulates in the air is between \(11.25 \mu g / m^3\) and \(12.75 \mu g / m^3\) with 99.7% confidence.
Hence, the correct answer is:
D. [tex]\(11.25 \mu g / m^3 - 12.75 \mu g / m^3\)[/tex]
1. Determine the necessary statistical parameters:
- Sample mean (\(\bar{x}\)): 12 \(\mu g / m^3\)
- Sample size (n): 196
- Sample standard deviation (s): 3.5 \(\mu g / m^3\)
- Confidence level: 99.7%
2. Convert the confidence level to a z-score:
- For a 99.7% confidence level, traditionally we use a z-score associated with this confidence level. In this case, the z-score is approximately 3.
3. Calculate the standard error of the mean (SE):
- The standard error is given by the formula:
[tex]\[ SE = \frac{s}{\sqrt{n}} \][/tex]
- Plugging in the values:
[tex]\[ SE = \frac{3.5}{\sqrt{196}} = \frac{3.5}{14} = 0.25 \][/tex]
4. Calculate the margin of error (ME):
- The margin of error is given by the formula:
[tex]\[ ME = z \times SE \][/tex]
- Plugging in the values:
[tex]\[ ME = 3 \times 0.25 = 0.75 \][/tex]
5. Determine the confidence interval:
- The confidence interval can be found using the formula:
[tex]\[ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) \][/tex]
- Plugging in the values:
[tex]\[ \text{Confidence Interval} = (12 - 0.75, 12 + 0.75) = (11.25, 12.75) \][/tex]
Therefore, the actual concentration of fine particulates in the air is between \(11.25 \mu g / m^3\) and \(12.75 \mu g / m^3\) with 99.7% confidence.
Hence, the correct answer is:
D. [tex]\(11.25 \mu g / m^3 - 12.75 \mu g / m^3\)[/tex]
