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Suppose that a random sample of 50 bottles of a particular brand of cough syrup 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 resulting \(95 \%\) confidence interval is \((7.8,9.4)\). a. Would a \(90 \%\) confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. 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: We can be highly confident that \(95 \%\) of all bottles of this type of cough syrup have an alcohol content that is between \(7.8\) and \(9.4\). Is this statement correct? Why or why not? d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding \(95 \%\) 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

Understanding Concept of Confidence Interval

A confidence interval provides a range of values that is likely to contain the population parameter with a certain level of confidence. A 95% confidence interval means that if we were to take many samples and create confidence intervals for each, we expect 95% of those intervals to contain the true population mean \(\mu\).

02

Examining Confidence Level Impact

When comparing confidence intervals with different confidence levels, a higher confidence level (e.g., 95%) produces a wider interval compared to a lower confidence level (e.g., 90%), given the same sample size. This is because a higher confidence level requires a larger margin of error to account for more uncertainty.

03

Evaluating 90% Interval Width Compared to 95% Interval

Since a 90% confidence interval requires less certainty than a 95% confidence interval, it results in a smaller margin of error and thus a narrower interval. Therefore, a 90% confidence interval would be narrower than the 95% confidence interval (7.8, 9.4).

04

Interpreting the 95% Confidence Interval

A 95% confidence interval means that we expect 95% of such calculated intervals from repeated samples to contain the true mean, \(\mu\). It does not mean that there is a 95% probability \(\mu\) is in our specific interval; \(\mu\) is a fixed parameter.

05

Explaining Population Proportion Misunderstanding

The confidence interval refers to the population mean, not the individual observations themselves. Thus, we are not stating that 95% of individual bottles fall within the interval; instead, we're discussing the mean alcohol content from repeated sample measures.

06

Understanding the Concept of Repeated Sampling

The statement that repeating the sampling and confidence interval calculation process 100 times will result in 95 intervals containing \(\mu\) aligns with the theory of confidence intervals, where 95% of intervals should contain the true mean in repeated sampling.

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Key Concepts

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

Population Mean

The population mean, denoted as \( \mu \), represents the average of a characteristic—in this case, alcohol content—across an entire population of interest, such as all bottles of a particular brand of cough syrup. It is a fixed, yet often unknown, value that researchers wish to estimate based on sample data. To find out the population mean, researchers select a sample. They then calculate the sample mean, which serves as a point estimate. However, this sample mean may not be exactly equal to the population mean due to natural variations in sampling.

To gain a better grasp of where the population mean lies, researchers use confidence intervals, which provide a range of values believed to contain the true mean. A key point to remember is that \( \mu \) is consistent; it is not something that changes or varies. The goal is to estimate it as accurately as possible using the data and statistical tools at our disposal.

Margin of Error

The margin of error is a crucial component when creating a confidence interval. It tells us how much the sample mean (our estimate) could vary from the actual population mean. Essentially, it provides a buffer zone around the sample mean, forming the confidence interval's boundaries.

A confidence interval can be expressed as:

• Lower Bound = Sample Mean - Margin of Error
• Upper Bound = Sample Mean + Margin of Error

The size of the margin of error depends on several factors, including the variation in the data and the confidence level chosen. A wider margin of error may mean more confidence in encompassing the true mean, yet it reduces precision. Similarly, a narrow margin suggests more precision but less certainty. Balancing these factors is key to estimating the population mean effectively.

Repeated Sampling

The concept of repeated sampling is fundamental in understanding confidence intervals. When we talk about a 95% confidence interval, we're essentially saying that if we repeatedly took samples from the same population and constructed confidence intervals for each, then about 95% of those intervals would capture the true population mean, \( \mu \).

This does not imply that within one specific interval, there is a 95% chance it contains \( \mu \). Instead, it is about long-term confidence in the process. Repeated sampling helps validate our interval estimates and gives us a general assurance about the reliability of our sample's proposed bounds. Through this, we align more closely with the actual population parameter.

Sample Size Impact

Sample size plays a pivotal role in constructing confidence intervals. Larger sample sizes tend to produce more precise confidence intervals because they provide a clearer picture of the population. As sample size increases:

• The variability or spread of our sample statistics decreases.
• The margin of error reduces, leading to narrower confidence intervals.

Why does this happen? It's because larger samples are more representative of the population, capturing more of the data's natural variation. Consequently, they lead to a more accurate estimate of the population mean and a smaller range of uncertainty. On the contrary, smaller samples might not capture the entire variability within the population, resulting in wider intervals and less reliable estimates.