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Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let \(\mu\) denote the mean alcohol content (in percent) for the population of all bottles of the brand under study. Suppose that the sample of 50 results in a \(95 \%\) confidence interval for \(\mu\) of \((7.8,9.4)\). a. Would a \(90 \%\) confidence interval have been narrower or wider than the given interval? Explain your answer. b. Consider the following statement: There is a \(95 \%\) chance that \(\mu\) is between \(7.8\) and \(9.4\). Is this statement correct? Why or why not? c. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding \(95 \%\) confidence interval is repeated 100 times, 95 of the resulting intervals will include \(\mu .\) Is this statement correct? Why or why not?

Step by step solution

01

Understand the confidence intervals

A confidence interval is an estimated range of possible values likely to include an unknown population parameter. The width of a confidence interval depends on the desired confidence level. A higher confidence level would result in a wider interval.

02

Respond to question a

For part (a), a \(90 \%\) confidence interval would have been narrower than a \(95 \% \) confidence interval. This is because a \(90 \%\) confidence interval embodies less uncertainty. A lesser confidence level is more specific (thus narrower), while a higher confidence level requires more range to ensure the actual parameter value falls within, hence wider.

03

Respond to question b

For part (b), the statement is incorrect. The \(95 \% \) confidence interval of \((7.8,9.4)\) implies that if we repeated our sample many times, \(95 \% \) of those samples would produce confidence intervals that contain the true population mean, \(\mu \). It does not mean that \(\mu \) has a \(95 \% \) chance of being between \(7.8 \) and \(9.4 \). The location of \(\mu \) is fixed, and the confidence interval is the range that we are \(95 \% \) confident contains that fixed \(\mu \).

04

Respond to question c

For part (c), the statement is correct. This describes the concept of a \(95 \% \) confidence interval correctly. If we repeated this process of taking samples and constructing confidence intervals 100 times, we would expect about \(95 \) of those intervals to contain the true population mean, \(\mu \). This is the essence of the meaning of a \(95 \% \) confidence interval.

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Key Concepts

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

Mean Alcohol Content

Understanding the concept of mean alcohol content is critical when examining substances like cough medicine. Essentially, the mean alcohol content is the average percentage of alcohol present in a sample set of bottles from a population. It is denoted by the symbol \(\mu\), which represents the true average across the entire population in this context.

When a study is conducted on cough medicine, researchers would assess a selection of bottles to estimate \(\mu\). This provides an insight into the overall alcohol content one might expect from any given bottle of that brand. In educational contexts, it's important to distinguish between the sample mean—a calculation from the chosen bottles—and the population mean, which is what the confidence interval aims to estimate.

Population Parameter Estimation

Population parameter estimation is a fundamental aspect of statistics. It involves inferring the value of a specific characteristic (a parameter) for a larger population based on sample data. In this case, the parameter of interest is the mean alcohol content in cough medicine bottles.

The aim is to use the sample data to estimate the population parameter \(\mu\) with a degree of confidence. Since it's impractical to measure every bottle, a sample provides a practical means to make an informed guess about the population. This guess, supported by statistical methods, gives us a range, known as the confidence interval, where we are reasonably sure the true mean resides.

Confidence Level and Interval Width

The relationship between confidence level and interval width is central to understanding confidence intervals. A confidence interval gives us a range within which we can be a certain percentage confident that the population parameter lies. The confidence level (e.g., 90%, 95%) indicates the degree of certainty we have in this assertion, while the interval width (the difference between the upper and lower bounds of the interval) shows the precision of our estimate.

If we desire more certainty about our estimate's accuracy (a higher confidence level), we must accept a wider interval. Conversely, a lower level of confidence allows us to claim a narrower interval but with less certainty that it contains the true parameter. It's a trade-off between certainty and precision that students should understand clearly.

Interpreting Confidence Intervals

Interpreting confidence intervals correctly is essential in statistics and often misunderstood. A \(95\%\) confidence interval, for instance, does not mean there is a \(95\%\) chance that the population parameter is within the given interval. Instead, it means that if we were to take an infinite number of samples and calculate a confidence interval from each, \(95\%\) of those intervals would be expected to contain the true population mean, \(\mu\).

In our cough medicine example, the interval does not move; it's our confidence in whether that fixed interval indeed includes the true \(\mu\) that the interval describes. By conducting repeated random sampling, one would find that approximately \(95\%\) of those intervals capture the true mean alcohol content—if the sampling and analysis are performed correctly.