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Answer :
To find a 95% confidence interval for the mean amount of caviar per tin, we'll follow these steps:
1. Identify the Given Information:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 99.8 grams
- Sample standard deviation (s) = 0.9 grams
- Sample size (n) = 20
2. Determine the Correct Distribution:
- Since the sample size is small (n < 30), we typically use the t-distribution to account for the extra uncertainty.
- We'll need the t-score that corresponds to a 95% confidence level with 19 degrees of freedom (n - 1 = 20 - 1).
3. Calculate the Standard Error (SE):
- The standard error of the mean is given by the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{0.9}{\sqrt{20}}
\][/tex]
4. Find the t-score for a 95% Confidence Interval:
- For a 95% confidence level with 19 degrees of freedom, the t-score is approximately 2.093.
5. Calculate the Margin of Error (ME):
- The margin of error is the t-score multiplied by the standard error:
[tex]\[
ME = 2.093 \times SE
\][/tex]
6. Construct the Confidence Interval:
- The confidence interval is found by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) = (99.8 - ME, 99.8 + ME)
\][/tex]
After carrying out these calculations, we find that Option B, which uses the t-score of 2.093, is the correct choice, as it provides the proper confidence interval for this scenario.
1. Identify the Given Information:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 99.8 grams
- Sample standard deviation (s) = 0.9 grams
- Sample size (n) = 20
2. Determine the Correct Distribution:
- Since the sample size is small (n < 30), we typically use the t-distribution to account for the extra uncertainty.
- We'll need the t-score that corresponds to a 95% confidence level with 19 degrees of freedom (n - 1 = 20 - 1).
3. Calculate the Standard Error (SE):
- The standard error of the mean is given by the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{0.9}{\sqrt{20}}
\][/tex]
4. Find the t-score for a 95% Confidence Interval:
- For a 95% confidence level with 19 degrees of freedom, the t-score is approximately 2.093.
5. Calculate the Margin of Error (ME):
- The margin of error is the t-score multiplied by the standard error:
[tex]\[
ME = 2.093 \times SE
\][/tex]
6. Construct the Confidence Interval:
- The confidence interval is found by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) = (99.8 - ME, 99.8 + ME)
\][/tex]
After carrying out these calculations, we find that Option B, which uses the t-score of 2.093, is the correct choice, as it provides the proper confidence interval for this scenario.
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