Correct The nutrition label for Oriental Spice Sauce states that one package of sauce has 1090 miligrams of sodium. To determine if the label is accurate, the FDA rand selects 50 packages of Oriental Spice Sauce and determines the sodium content. The sample has an average of $1001.30 \mathrm{milligrams}$ of sodium per package with standard deviation of 256.48 milligrams. Step 1 of 2: Calculate a $95 \%$ confidence interval for the mean sodlum content in Oriental Spice Sauce. Round your answers to two decimal places. Answer Tables
Solution
Step 1 ：The nutrition label for Oriental Spice Sauce states that one package of sauce has 1090 milligrams of sodium. To determine if the label is accurate, the FDA randomly selects 50 packages of Oriental Spice Sauce and determines the sodium content. The sample has an average of 1001.30 milligrams of sodium per package with a standard deviation of 256.48 milligrams.
Step 2 ：We need to calculate a 95% confidence interval for the mean sodium content in Oriental Spice Sauce. The formula for a confidence interval is: \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score which corresponds to the desired confidence level (for a 95% confidence level, the z-score is approximately 1.96), \(s\) is the sample standard deviation, and \(n\) is the sample size.
Step 3 ：Substituting the given values into the formula, we have: \(\bar{x} = 1001.30\) milligrams, \(z = 1.96\), \(s = 256.48\) milligrams, and \(n = 50\).
Step 4 ：Calculating the margin of error, we get approximately 71.09 milligrams.
Step 5 ：Subtracting the margin of error from the sample mean, we get the lower limit of the confidence interval, which is approximately 930.21 milligrams.
Step 6 ：Adding the margin of error to the sample mean, we get the upper limit of the confidence interval, which is approximately 1072.39 milligrams.
Step 7 ：This means that we are 95% confident that the true mean sodium content in Oriental Spice Sauce is between 930.21 milligrams and 1072.39 milligrams.
Step 8 ：Final Answer: The 95% confidence interval for the mean sodium content in Oriental Spice Sauce is approximately \(\boxed{[930.21, 1072.39]}\) milligrams.
