Question

Suppose that a random sample of 50 bottles of a par. ticular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let \(\mu\) denote the average alcohol content 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?

Short answer

a. A \(90 \%\) would have been narrower than the \(95 \%\) interval. b. It is incorrect to say there's a \(95 \%\) chance the average alcohol content is between \(7.8\) and \(9.4\); the correct interpretation is that we're \(95 \%\) confident the actual average falls within our interval. c. It’s incorrect to say that \(95\) out of \(100\) confidence intervals will contain the average; the correct interpretation is that, in the long run, \(95 \%\) of the intervals resulting from this process would contain the average.

Step-by-step solution

Understanding and comparing confidence intervals

Part (a) asks whether a \(90 \%\) confidence interval would be wider or narrower than a \(95 \%\). Because a higher confidence level corresponds to a wider interval - given that a higher confidence level means more certainty about the parameter falling within the interval - a \(90 \%\) confidence interval would be narrower than the \(95 \%\).

Interpreting the confidence interval

Part (b) asks whether it is correct to say there's a \(95 \%\) chance that the average alcohol content is between \(7.8\) and \(9.4\). The correct interpretation of a \(95 \%\) confidence interval is that we're \(95 \%\) confident that the actual average falls within our interval. The average alcohol content isn't a random variable - it's a fixed value - we're just not exactly sure what it is. So, it's incorrect to say there's a 'chance' it falls within the interval.

Understanding the concept of 'confidence' in confidence intervals

Part (c) asks whether it is correct to say if sampling and confidence interval calculation is repeated \(100\) times, \(95\) intervals will include the actual average. It’s important to note that 'confidence' in confidence intervals doesn’t mean that \(95 \%\) of the samples will always contain the population parameter but that, in the long run, \(95 \%\) of such computed intervals would contain the population parameter. The emphasis is on the method rather than the individual intervals which are outcomes of the method.

Key concepts

Statistical Inference

Statistical inference is a cornerstone of data analysis, allowing us to draw conclusions about a population based on sample data. It involves using probability theory to estimate population characteristics from samples. When we infer the average alcohol content of cough medicine bottles from a sample of 50, we're applying statistical inference.

Understanding confidence intervals, as in the provided exercise, is fundamental to statistical inference. Given a 95% confidence interval of (7.8, 9.4), we infer there's a high level of certainty that the true population mean, \( \mu \), lies within this range. However, it's crucial to grasp that the 'confidence' is not about the probability of \( \mu \) being in the interval but about the reliability of our estimation process if repeated over time.

Population Parameter Estimation

Population parameter estimation involves deducing the value of a particular characteristic (parameter) for an entire population. In our exercise, this parameter is the mean alcohol content \( \mu \). Estimation is performed through samples because studying the entire population is often not feasible.

This estimation process arises from the collected sample's data, leading us to calculate a range where we believe the true population parameter will lie. A narrower confidence interval, such as a hypothetical 90% interval compared to a 95% interval, indicates more precision but less confidence. It's like saying, 'We're taking a calculated gamble and are less sure, but we think the true mean is closer to this specific number.'

Sample Data Analysis

Sample data analysis is the process by which we examine and interpret the data collected from a subset of the population (our sample) to make inferences about the whole. This method assumes that the sample is representative of the population and applies statistical techniques to derive meaningful patterns and measures, such as confidence intervals.

In part (c) of our exercise, the interpretation of the sample analysis process is crucial. The statement isn't asserting that exactly 95 of the 100 intervals will include \( \mu \); rather, it expresses confidence in the long-term performance of the method. In other words, if we kept taking samples and calculating intervals, we'd expect that 95% of those intervals would capture the true mean—demonstrating the powerful insights we can gain from well-conducted sample data analysis.