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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 for the population of all bottles of this brand. Suppose that this 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 random sample of size 50 and then calculating the corresponding \(95 \%\) confidence interval is repeated 100 times, exactly 95 of the resulting intervals will include \(\mu .\) Is this statement correct? Why or why not?

Step by step solution

01

Understanding Confidence Level Impact on Interval Width

When you decrease the confidence level, from \(95%\) to \(90%\) for instance, the width of the confidence interval decreases as well. The confidence level is a measure of certainty. If we are less certain (\(90%\) compared to \(95%)\), we need less precision, hence a narrower confidence interval.

02

Identifying Errors in Confidence Interval Interpretation

The statement 'There is a \(95%\) chance that the mean alcohol content \(\mu\) is between 7.8 and 9.4' is incorrect. Confidence intervals do not work that way. The correct interpretation should be: We are \(95%\) confident that the interval (7.8, 9.4) contains the true mean alcohol content \(\mu\). It is not about probability of \(\mu\), but about the level of confidence in our method.

03

Understanding Probabilistic Interpretation of Confidence Intervals

Similarly, the statement 'If the process is repeated 100 times, exactly 95 of the intervals will include \(\mu\)' is incorrect as well. This is a common misunderstanding. What the \(95%\) confidence level means is that \(95%\) of random samples of the same size from the same population will result in confidence intervals that contain the population parameter. However, it does not state any guarantee about the exact number out of 100.

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Key Concepts

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

Confidence Level

When you hear about the "confidence level," think of it as the degree of certainty or assurance we have in our statistical method. In the context of confidence intervals, a common measure of confidence is expressed as a percentage, such as 90%, 95%, or 99%. These percentages represent how sure we are that the method used to create our interval will capture the true population parameter, like a mean.

For instance, a 95% confidence level suggests that if we repeatedly took random samples and calculated the confidence intervals, approximately 95% of these intervals would successfully contain the true parameter we are trying to estimate. Thus, the confidence level is not about a single interval being correct or not, but rather reflects the reliability of the process used to create those intervals.

Population Mean

In statistics, the population mean (\(\mu\),) is a crucial concept. It represents the average value of a given attribute for an entire population. Unlike the sample mean, which is computed from a subset, the population mean is about the entire group we're analyzing or interested in.

For example, in our exercise, \(\mu\) could stand for the average alcohol content in all bottles of a specific brand of cough medicine. Knowing the population mean helps us understand the overall characteristic of the population, even if it's not always directly observable.

Estimation techniques, like confidence intervals, are designed to help us infer or predict the population mean based on data from a limited sample, which leads us to a better understanding of the entire population.

Random Sample

A random sample is a fundamental idea in statistics that involves selecting a subset of individuals from a population in such a way that each subset has an equally likely chance of being chosen. This is important because it helps ensure that the sample truly represents the population, free from biases.

In our scenario, taking a random sample of 50 bottles of cough medicine means that every group of 50 bottles from all possible bottles has the same chance of being selected. This randomness is critical because it provides the foundation for making valid statistical inferences, like calculating confidence intervals which can estimate the population mean from the subset data.

Interval Width

The concept of interval width deals with how broad or narrow a confidence interval is. This width indicates the range of values we can be confident contains the true population parameter. Several factors influence the interval's width, including the sample size and the confidence level.

Generally speaking, if we decrease our confidence level (say from 95% to 90%), the interval will get narrower because we are essentially accepting more uncertainty about whether the interval contains the true population mean. Conversely, to be more confident (e.g., 99%), we widen the interval to increase the likelihood it holds the true parameter.

In addition, a larger sample size tends to make intervals narrower because it provides more information and thus greater precision about what the population is truly like.

Statistical Inference

Statistical inference is about drawing conclusions from data. At its heart, it involves determining how likely it is that a pattern or effect observed in a sample also exists in the overall population. This process relies heavily on concepts like random sampling and constructing confidence intervals.

Through statistical inference, we use sample data to make educated statements or predictions about the larger population. For example, estimating the population mean from a sample through a confidence interval allows us to infer its value with a stated level of confidence.

Statistical inference helps turn raw data into meaningful insights, which is why it's a cornerstone of data analysis and decision-making in diverse fields, from medicine to market research.