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Suppose that a random sample of 50 cans of a particular brand of fruit juice is selected, and the amount of juice (in ounces) in each of the cans is determined. Let \(\mu\) denote the mean amount of juice for the population of all cans 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

a. Comparison of Confidence Intervals

A 90% confidence interval would be narrower than the given 95% confidence interval. Confidence intervals are determined by their level of confidence. A higher confidence level results in a wider interval because there is more uncertainty. In this case, having a 95% confidence means that there is higher confidence that the true mean is within this interval, resulting in the interval's width being larger than that of the 90% interval.

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b. Assessing the Statement

The statement "There is a 95% chance that μ is between 7.8 and 9.4" is not correct. The concept of a confidence interval is often misunderstood. When we have a 95% confidence interval, it means that if we would repeat the process of constructing an interval again and again based on new random samples, 95% of these intervals would contain the true population mean. In other words, the 95% confidence is a property of the procedure, not of the specific interval. The true mean μ is either within this interval, or it is not.

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c. Evaluating the Statement

The 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 μ" is incorrect. The concept of a 95% confidence interval means that approximately 95% of the resulting intervals will contain the true mean, not exactly 95 of them. However, it is important to note that this interpretation of confidence intervals is based on a long-run relative frequency. It doesn't mean that in every set of 100 intervals exactly, 95 will contain the true mean, but it gives an indication of how often we can expect the intervals to include the true mean in the long run.

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Key Concepts

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

Confidence Level

When we talk about a confidence level, we're referring to how "confident" we are in our interval estimate of a population parameter. For example, the "95% confidence level" indicates that if we drew many random samples and constructed a confidence interval from each sample, we would expect about 95% of these intervals to contain the true population mean.
This means a higher confidence level leads to a wider confidence interval because we need to encompass more to be more certain that the true mean lies within our estimated range. Conversely, a lower confidence level would result in a narrower interval.

• A 95% confidence level gives a wider interval than a 90% confidence level because we are more certain about the estimate.
• This level does not guarantee that any specific interval contains the true mean; it speaks to the reliability of the process over many samples.

Understanding this helps correct the common misconception that a 95% confidence interval provides a 95% probability that the population mean is within that specific interval, which is not accurate.

Population Mean

The population mean, represented by \( \mu \), is the average of a particular measurement across an entire population. In our exercise, it is the average amount of fruit juice in all the cans of the specified brand. The goal of calculating a confidence interval is to estimate this unknown population mean based on the sample mean.
When we calculate the confidence interval, we use the sample data as a basis to infer about the entire population. This allows us to make educated guesses about the population mean. Here's why the population mean is critical:

• The sample mean provides an approximation of the population mean.
• We use statistical methods to estimate how much the sample mean might differ from the actual population mean.

This concept underlies the importance of the confidence interval, as it gives us a range within which we expect the true population mean to fall, based on the sample data.

Random Sample Selection

Random sample selection is crucial in statistical procedures because it ensures that every individual of the population has an equal chance of being included in the sample. This randomness helps prevent bias in the results, making the sample more representative of the population.
In the exercise scenario, the random sample of 50 cans was selected to examine the amount of juice in each. The purpose of random sampling is:

• To improve the accuracy of the sample mean as an estimator for the population mean.
• To increase the validity and reliability of the confidence interval produced.

Random sampling allows statisticians to confidently project the findings from the sample to the entire population. Without random selection, the results obtained from the sample could be skewed or biased, leading to inaccurate estimates of the population mean and confidence interval.