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Chapter 8: Problem 78

Multiple choice: Select the best answer for Exercises 75 to 78. Scientists collect data on the blood cholesterol levels (milligrams per deciliter of blood) of a random sample of 24 laboratory rats. \({A} 95 \%\) confidence interval for the mean blood cholesterol level \(\mu\) is 80.2 to 89.8 . Which of the following would cause the most worry about the validity of this interval? (a) There is a clear outlier in the data. (b) A stemplot of the data shows a mild right-skew. (c) You do not know the population standard deviation \(\sigma\) (d) The population distribution is not exactly Normal. (e) None of these would be a problem because the t procedures are robust.

Short Answer

Expert verified
(a) There is a clear outlier in the data.

Step by step solution

01

Understand Confidence Interval Validity

The 95% confidence interval provides an estimated range of values which is likely to include the true mean blood cholesterol level of the laboratory rats. This interval is based on the t-distribution, which is typically used due to the unknown population standard deviation and small sample size.
02

Analyze Impact of an Outlier

An outlier can significantly affect the mean and confidence interval, potentially distorting the results. In small sample sizes like this one (n=24), an outlier can have a large impact on the data.
03

Consider Data Skewness

A mild right skew might slightly affect the symmetry assumed by the t-distribution, but it is not typically a major concern if the skew is mild, given the robustness of t-procedures, especially if the sample size is moderately large.
04

Evaluate Unknown \\(\sigma\\)

The population standard deviation \(\sigma\) is indeed unknown, which is why the t-distribution is used instead of the z-distribution. This does not invalidate the confidence interval itself as t-procedures are designed for this purpose.
05

Assess Population Distribution Normality

While the population distribution isn't exactly normal, t-procedures are robust to deviations from normality, particularly if the sample size is large. With a sample size of 24, slight deviations from normality are typically not concerning.
06

Determine Which Cause is Most Concerning

Among the options, a clear outlier (option a) would cause the most worry about the interval's validity because it can heavily influence the interval estimation.

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Key Concepts

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

T-distribution
The t-distribution is a powerful statistical tool. It helps estimate population parameters when the sample size is small and the population standard deviation is unknown. Unlike the normal distribution, the t-distribution is wider and has heavier tails. This means it accounts for more variability, which is crucial in these situations. Key points about the t-distribution:
  • It's used when the sample size is small (usually n < 30).
  • It's appropriate when the population standard deviation is not known.
  • The degrees of freedom (df) are related to the sample size, with df = n - 1.
In our case with the laboratory rats, a sample size of 24 means we're right at the edge of what is considered a small sample. Thus, using the t-distribution remains appropriate, particularly appreciating its robust nature in handling slight variations from normality.
Outliers in Data
Outliers are extreme values that deviate significantly from the rest of the dataset. They can dramatically affect statistical analyses by skewing the results. In the context of confidence intervals, outliers play an important role. Here's how outliers impact confidence interval validity:
  • They can skew the mean, potentially resulting in a misleading confidence interval range.
  • With a small sample size, their impact is magnified.
While the t-distribution accounts for some variability, a single outlier could still introduce significant distortion. That's why having a clear outlier in our data about lab rats' cholesterol levels is worrisome. It can lead to incorrect conclusions about the population mean, and thus, it remains the most concerning factor in maintaining the interval's validity.
Sample Size Impact
Sample size is a crucial element when forming confidence intervals. It influences the precision and accuracy of the estimate. The larger the sample size, the more stable and reliable the confidence interval becomes. Considerations with sample size impact:
  • Smaller sample sizes yield less precise confidence intervals, which means a wider interval range.
  • Larger samples provide estimates closer to the true population parameter.
In the case of our 24 rat samples, while it fits within the general framework for t-distribution use, it's right on the edge of what's considered small. Hence, the sample size itself doesn't invalidate the interval, but care should be taken to ensure that the results are interpreted with the awareness of potential variability due to its size.
Robust Statistics
Robust statistics are designed to perform well under violations of common assumptions, such as the normality of distribution and the presence of outliers. T-procedures are a perfect example of robust statistics. Understanding robust statistics:
  • They are less sensitive to outliers and departures from assumed distributions.
  • They provide meaningful results even when assumptions like normality are slightly violated.
In our exercise, using t-procedures illustrates how robust these methods are. They allow us to still glean valuable insights from the data, like the 95% confidence interval for blood cholesterol levels, even if the population distribution isn't perfectly normal or if data has mild skewness. This robustness prevents minor assumption breaks from significantly skewing the final results, provided outliers are carefully managed.

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