Questions: A researcher randomly selects and measures the volumes of the contents of 15 bottles of cough syrup. Assume the sample is taken from a normally distributed population. The 98% confidence interval for the population standard deviation σ is (0.0283,0.0456). The population standard deviation of the volumes of the bottles' contents should be less than 0.025 fluid ounce. Does the confidence interval for σ suggest that the variation in the volumes of the bottles' contents is at an acceptable level? Explain your reasoning. Choose the correct answer below. A. Yes, because all values contained in the confidence interval are less than 0.025. B. Yes, because 0.025 is contained in the confidence interval. C. Yes, because all values contained in the confidence interval are greater than 0.025. D. No, because all values contained in the confidence interval are less than 0.025. E. No, because all values contained in the confidence interval are greater than 0.025. F. No, because 0.025 is contained in the confidence interval.

A researcher randomly selects and measures the volumes of the contents of 15 bottles of cough syrup. Assume the sample is taken from a normally distributed population. The 98% confidence interval for the population standard deviation σ is (0.0283,0.0456). The population standard deviation of the volumes of the bottles' contents should be less than 0.025 fluid ounce. Does the confidence interval for σ suggest that the variation in the volumes of the bottles' contents is at an acceptable level? Explain your reasoning. Choose the correct answer below. A. Yes, because all values contained in the confidence interval are less than 0.025. B. Yes, because 0.025 is contained in the confidence interval. C. Yes, because all values contained in the confidence interval are greater than 0.025. D. No, because all values contained in the confidence interval are less than 0.025. E. No, because all values contained in the confidence interval are greater than 0.025. F. No, because 0.025 is contained in the confidence interval.
Transcript text: A researcher randomly selects and measures the volumes of the contents of 15 bottles of cough syrup. Assume the sample is taken from a normally distributed population. The $98 \%$ confidence interval for the population standard deviation $\sigma$ is $(0.0283,0.0456)$. The population standard deviation of the volumes of the bottles' contents should be less than 0.025 fluid ounce. Does the confidence interval for $\sigma$ suggest that the variation in the volumes of the bottles' contents is at an acceptable level? Explain your reasoning. Choose the correct answer below. A. Yes, because all values contained in the confidence interval are less than 0.025 . B. Yes, because 0.025 is contained in the confidence interval. C. Yes, because all values contained in the confidence interval are greater than 0.025 . D. No, because all values contained in the confidence interval are less than 0.025 . E. No, because all values contained in the confidence interval are greater than 0.025 . F. No, because 0.025 is contained in the confidence interval.
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Solution

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Solution Steps

Step 1: Confidence Interval Analysis

The confidence interval for the population standard deviation \( \sigma \) is given as \( (0.0283, 0.0456) \). This means that we are \( 98\% \) confident that the true population standard deviation lies within this range.

Step 2: Acceptable Standard Deviation

The acceptable standard deviation for the volumes of the bottles' contents is specified as \( 0.025 \) fluid ounces.

Step 3: Comparison of Bounds

We compare the bounds of the confidence interval with the acceptable standard deviation:

  • Lower bound: \( 0.0283 \)
  • Upper bound: \( 0.0456 \)
  • Acceptable standard deviation: \( 0.025 \)
Step 4: Conclusion

Since both bounds of the confidence interval are greater than \( 0.025 \): \[ 0.0283 > 0.025 \quad \text{and} \quad 0.0456 > 0.025 \] This indicates that the variation in the volumes of the bottles' contents is not at an acceptable level.

Final Answer

The answer is \( \boxed{E} \). No, because all values contained in the confidence interval are greater than \( 0.025 \).

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