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Chapter 17: Problem 38

Hot dogs A nutrition lab tested 40 hot dogs to see if their mean sodium content was less than the \(325-\mathrm{mg}\) upper limit set by regulations for "reduced sodium" franks. A \(90 \%\) confidence interval estimated the mean sodium content for this kind of hot dog at 317.2 to \(326.8 \mathrm{mg}\). Given this, what would you tell the lab about whether the hot dogs satisfy the regulation.

Short Answer

Expert verified
Even though the majority of the time (specifically \(90\%\)) the sodium content will be within the stipulated limit, there is a \(5\%\) chance that the sodium content could exceed the limit. Thus, we cannot conclusively declare that these hot dogs meet the 'reduced sodium' regulations.

Step by step solution

01

Understanding Confidence Interval

First, it's important to remember what a sustainability confidence interval means in statistics. A \(90\%\) confidence interval for the mean implies that, if we repeated the test many times, \(90\%\) of such intervals would include the true mean sodium content of the hot dogs.
02

Analyzing the Confidence Interval

Now let's look at the given confidence interval, which goes from \(317.2mg\) to \(326.8mg\). This implies that we can be \(90\%\) confident that the true mean sodium content lies somewhere in this interval.
03

Comparing the Confidence Interval with the Regulation Limit

The regulation states that to be classified as 'reduced sodium', the sodium content must be less than \(325mg\). However, our confidence interval goes as high as \(326.8mg\), meaning that in \(5\%\) of tests (since it exceeds the regulation at the upper end of the interval, and this end represents \(5\%\) of the \(90\%\) interval), the mean sodium content is likely to be greater than \(325mg\).
04

Drawing a Conclusion

Since there is a possibility (even though it's just \(5\%\)) that the mean sodium content could exceed the regulation limit, it would be statistically incorrect to unequivocally declare that these hot dogs satisfy the regulation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Hypothesis Testing
Statistical hypothesis testing is a method used to determine if there is enough statistical evidence in a sample of data to infer that a certain condition is true for the entire population. In the context of product testing, like with our hot dog example, hypothesis testing might be used to establish whether the product complies with certain regulations, such as the sodium content in

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Most popular questions from this chapter

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