91Ó°ÊÓ

Multiple choice: Number of close friends \(\quad\) Based on responses of 1467 subjects in a General Social Survey, a \(95 \%\) confidence interval for the mean number of close friends equals \((6.8,8.0) .\) Which \(t w o\) of the following interpretations are correct? a. We can be \(95 \%\) confident that \(\bar{x}\) is between 6.8 and 8.0 . b. We can be \(95 \%\) confident that \(\mu\) is between 6.8 and 8.0 . c. Ninety-five percent of the values of \(X=\) number of close friends (for this sample) are between 6.8 and \(8.0 .\) d. If random samples of size 1467 were repeatedly selected, then \(95 \%\) of the time \(\bar{x}\) would be between 6.8 and \(8.0 .\) e. If random samples of size 1467 were repeatedly selected, then in the long run \(95 \%\) of the confidence intervals formed would contain the true value of \(\mu\).

Step by step solution

01

Understanding Confidence Interval

The confidence interval for the mean number of close friends is given as \((6.8, 8.0)\). This interval is created from a sample to estimate the population parameter \(\mu\), the true mean number of close friends.

02

Analyzing Option (a)

Option (a) states: "We can be \(95\%\) confident that \(\bar{x}\) is between 6.8 and 8.0." The confidence interval is for the population mean \(\mu\), not the sample mean \(\bar{x}\). Therefore, option (a) is incorrect.

03

Analyzing Option (b)

Option (b) states: "We can be \(95\%\) confident that \(\mu\) is between 6.8 and 8.0." This is a correct interpretation of a confidence interval, which estimates the range within which the true population mean \(\mu\) is likely to be.

04

Analyzing Option (c)

Option (c) states: "Ninety-five percent of the values of \(X=\) number of close friends (for this sample) are between 6.8 and \(8.0\)." The confidence interval does not refer to the spread of individual data values, but rather to the estimation of the mean. Thus, option (c) is incorrect.

05

Analyzing Option (d)

Option (d) states: "If random samples of size 1467 were repeatedly selected, then \(95\%\) of the time \(\bar{x}\) would be between 6.8 and \(8.0\)." This misinterprets the concept of a confidence interval, which does not predict individual sample means but whether they contain \(\mu\). Thus, option (d) is incorrect.

06

Analyzing Option (e)

Option (e) states: "If random samples of size 1467 were repeatedly selected, then in the long run \(95\%\) of the confidence intervals formed would contain the true value of \(\mu\)." This is a correct interpretation of what it means to have a \(95\%\) confidence interval.

07

Conclusion

The interpretations that are correct are options (b) and (e). These accurately describe the purpose and function of a \(95\%\) confidence interval, which is to estimate the population mean \(\mu\) and state the long-term behavior of the intervals.

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Key Concepts

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

Population Parameter

In the world of statistics, a population parameter is a numerical value that defines a characteristic of an entire population. In this context, the population is all possible subjects that could be surveyed for their number of close friends. The population parameter often involves values such as the population mean, denoted as \( \mu \). When we calculate a confidence interval, we're trying to estimate this true population mean \( \mu \). The goal is to understand where \( \mu \) likely falls, based on data from a sample. Since it's usually impractical to gather data from an entire population, confidence intervals offer insight by providing a range where \( \mu \) might be, giving researchers useful inferential power.

Sample Mean

The sample mean, denoted as \( \bar{x} \), is the average value obtained from a sample of the population. In our exercise, the sample consists of 1467 individuals. This sample mean is calculated by adding up all the data points (the number of close friends each subject has) and dividing by the number of data points (1467 in this case).Here are a few key points to know about the sample mean:

• It serves as an estimate of the population mean \( \mu \).
• It is used to construct the confidence intervals.

However, it is important to remember that the sample mean is just an estimate and might not perfectly reflect the population mean. This is one reason confidence intervals are crucial for proper statistical understanding.

Statistical Interpretation

Statistical interpretation is about understanding what the results of an analysis mean. In this exercise, the statistical interpretation revolves around the confidence interval \( (6.8, 8.0) \) and its relationship to the population mean \( \mu \).When we say there is a \(95\%\) confidence interval of \( (6.8, 8.0) \), we're stating that we are confident the true mean number of close friends lies within this range. This does not mean that \(95\%\) of individual sample means \( \bar{x} \) fall in this range, nor does it mean 95\% of individual values fall within this interval.The key takeaway for proper statistical interpretation is in the understanding of what confidence intervals truly represent — an estimated range for the population parameter \( \mu \).

Confidence Level

A confidence level is a measure of how frequently the true population parameter is expected to be within the confidence interval if you repeated the sampling process multiple times. In our exercise, the confidence level is \(95\%\).This means that, if you were to take 100 different samples and build a confidence interval from each of them, roughly 95 of those intervals would contain the actual population mean \( \mu \). It is a reflection of the reliability of the estimation process:

• Higher confidence levels require wider intervals as they need to capture more of the possible values for \( \mu \).
• The confidence level is always chosen before analyzing data and reflects the acceptable risk of making an incorrect inference.

Understanding this concept aids in appreciating why certain options in our multiple-choice problem are correct while others aren't.