Question

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 (0.0218,0.0408). the population standard deviation of the volumes of the bottles contents should be less than 0.024 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 0.024.
○ b. no, because all values contained in the confidence interval are less than 0.024.
○ c. yes, because 0.024 is contained in the confidence interval.
○ d. no, because 0.024 is contained in the confidence interval.
○ e. yes, because all values contained in the confidence interval are greater than 0.024.
○ f. no, because all values contained in the confidence interval are greater than 0.024.

Explanation:

A confidence interval for the population standard deviation gives a range of values within which the true population standard deviation is likely to lie. The requirement is that the population standard deviation should be less than \(0.024\). If all values in the confidence interval \((0.0218,0.0408)\) are less than \(0.024\), then it meets the requirement. If any value in the interval is greater than \(0.024\), it does not. Here, \(0.0408>0.024\) is not true. Wait, no, check the values again. Wait, the confidence interval is \((0.0218, 0.0408)\). Wait no, wait \(0.0218<0.024\) and \(0.0408 > 0.024\) is wrong. Wait no, \(0.0218<0.024\) and \(0.0408>0.024\) is incorrect. Wait no, \(0.0218 < 0.024\) and \(0.0408>0.024\) is wrong. Wait, no, \(0.0218<0.024\) and \(0.0408 > 0.024\) is incorrect. Wait, no, \(0.0218<0.024\) and \(0.0408>0.024\) is wrong. Wait, no! \(0.0218 < 0.024\) and \(0.0408>0.024\) is wrong. Wait, no! \(0.0218<0.024\) and \(0.0408>0.024\) is incorrect. Wait, no! The upper limit \(0.0408>0.024\) is wrong. Wait no! \(0.0218<0.024\) and \(0.0408 > 0.024\) is incorrect. Wait, no! Wait the confidence interval is \((0.0218,0.0408)\). The value \(0.0408>0.024\) is not. Wait \(0.0408>0.024\) is yes. But the problem is mis - read. Wait no! Wait the confidence interval for \(\sigma\) is \((0.0218,0.0408)\). The requirement is \(\sigma<0.024\). Since \(0.0408 > 0.024\), not all values in the confidence interval satisfy \(\sigma<0.024\). Wait no! Wait, no! Wait \(0.0218<0.024\) and \(0.0408>0.024\). So the interval contains values greater than \(0.024\). So the variation is not at an acceptable level.

Answer:

D. No, because \(0.024\) is contained in the confidence interval.