A company producing over 1,000 tins of caviar per day took a simple random sample (SRS) of 20 tins from one day's production. The sample showed a mean of 99.8 g of caviar per tin with a standard deviation of 0.9 g. The amounts were roughly symmetric with no outliers.

Based on this sample, which of the following is a [tex]$95\%$[/tex] confidence interval for the mean amount of caviar (in grams) per tin from that day's production?

Choose one answer:

A. [tex]$99.8 \pm 1.96\left(\frac{0.9}{\sqrt{20}}\right)$[/tex]

B. [tex]$99.8 \pm 2.093\left(\frac{0.9}{\sqrt{20}}\right)$[/tex]

C. [tex]$99.8 \pm 1.96 \sqrt{\frac{0.9(0.1)}{20}}$[/tex]

D. [tex]$99.8 \pm 1.96(0.9)$[/tex]

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.