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More conclusions In January \(2002,\) two students made worldwide headlines by spinning a Belgian euro 250 times and getting 140 heads - that's \(56 \% .\) That makes the \(90 \%\) confidence interval \((51 \%, 61 \%) .\) What does this mean? Are these conclusions correct? Explain. a) Between \(51 \%\) and \(61 \%\) of all euros are unfair. b) We are \(90 \%\) sure that in this experiment this euro landed heads on between \(51 \%\) and \(61 \%\) of the spins. c) We are \(90 \%\) sure that spun euros will land heads between \(51 \%\) and \(61 \%\) of the time. d) If you spin a curo many times, you can be \(90 \%\) sure of getting between \(51 \%\) and \(61 \%\) heads. e) \(90 \%\) of all spun euros will land heads between \(51 \%\) and \(61 \%\) of the time.

Step by step solution

01

Understanding Confidence Intervals

A confidence interval provides a range within which we expect the true population parameter to lie, given a certain level of confidence, usually represented as a percentage (in this case, 90%). It is important to note that it does not guarantee that the true parameter falls within this range but rather that if we were to take many samples, 90% of the calculated intervals would contain the true parameter.

02

Analyzing Option (a)

Statement (a) says that between 51% and 61% of all euros are unfair. This statement incorrectly interprets the confidence interval as stating the proportion of unfair euros themselves, which is not what a confidence interval means.

03

Analyzing Option (b)

Statement (b) suggests we are 90% sure that in this single experiment, the euro landed on heads between 51% and 61% of the time. This is an incorrect interpretation, as the confidence interval provides information about the true proportion of heads over many experiments, not repetitive outcomes of a single experiment.

04

Analyzing Option (c)

Statement (c) suggests we are 90% confident that euros will land heads between 51% and 61% of the time across multiple spins. This correctly interprets the confidence interval as about the true proportion in repeated experiments, aligning with the purpose of confidence intervals.

05

Analyzing Option (d)

Option (d) suggests that spinning a euro multiple times ensures a 90% guarantee of getting heads between 51% and 61%. This is incorrect as it misinterprets the confidence interval as predicting outcomes of repeated spins in a deterministic way rather than probabilities over repeated sampling.

06

Analyzing Option (e)

Statement (e) claims that 90% of all spun euros will land on heads between 51% and 61% of the time. This interprets the interval incorrectly as if it were about individual euro outcomes, which overgeneralizes the result of a single experiment to population characteristics.

07

Conclusion

The correct interpretation of the 90% confidence interval is provided by option (c). It states that we are 90% confident that the true proportion of heads lies between 51% and 61% for the euro based on repeated sampling.

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Key Concepts

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

Statistical Interpretation

Confidence intervals play a crucial role in statistical interpretation. They help us understand the range within which a true population parameter might fall based on our sample data.
In our example, the confidence interval of 51% to 61% is about the true proportion of how often a euro lands heads when spun. It is important to grasp that this interval doesn't provide certainty. Instead, it offers a degree of confidence - in our case, 90% - that the true parameter is captured within this range when considering repeated studies.

• This is not to be interpreted as saying that all euros are unfair.
• The interval does not predict results of a single experiment.
• It is about the possible true proportion over many similar experiments.

In simple terms, if you repeated the experiment multiple times, 90% of those experiments would have their confidence intervals covering the true proportion of heads.

Probability

Probability is a foundational concept when discussing confidence intervals. It helps express the level of confidence associated with the interval.
The 90% in a 90% confidence interval speaks to probability. It refers to how sure we are that this interval will enclose the true parameter upon repeated sampling. However, it is not a probability estimate about individual events or outcomes of a single spin.

• Probability guides us in understanding how often our interval would contain the truth.
• It doesn't claim that outcomes within the interval will occur 90% of the time.

This aspect of probability in statistics highlights the difference between chance outcomes in individual trials versus trends observed across many experiments.

Sample Size

Sample size is crucial when constructing confidence intervals. It influences the width of the interval, which reflects our estimate's precision.
For instance, in our example using 250 euro spins, the sample size of 250 is a substantial number. It allows for a more reliable estimation of the interval than a smaller sample would. Generally, larger samples provide more precise estimates.

• A large sample size leads to more accurate interval estimations.
• Smaller samples might result in wider intervals, indicating less precision.
• Thus, recognizing the role of sample size helps in evaluating the robustness of statistical findings.

The size of the sample used is vital. It influences how confident we can be about the conclusions drawn from our study.

Experimental Study

In an experimental study, like spinning a euro multiple times, researchers collect data to draw conclusions about a population parameter.
This involves conducting experiments repeatedly to ensure the findings are reliable and not due to chance. Here, the experiment of spinning a euro 250 times allows us to make statistical inferences about the likelihood of the euro landing on heads.

• The repeated nature of experiments strengthens the validity of statistical inferences.
• Experimental studies often seek to control variables to draw clearer conclusions.

The integrity of our experimental study depends on careful design and execution, which in turn determines how confidently we can assert findings based on our confidence intervals.