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 w is 80.2 to 89.8.
Which of the following would cause the most worry about the validity of this interval?
There is a clear outlier in the data.
None of these are a problem when using a r interval
O A stemplot of the data shows a mild right skew.
You do not know the population standard deviation
o The population distribution is not exactly Normal
The correct answer and explanation is:
The correct answer is:
“There is a clear outlier in the data.”
Explanation:
Confidence intervals rely on certain assumptions to be valid, particularly when estimating a population mean. The primary assumption is that the sample comes from a normally distributed population or that the sample size is large enough for the Central Limit Theorem (CLT) to apply. However, certain factors can compromise the accuracy of the interval, with outliers being one of the most concerning.
- Why Outliers Are Problematic?
Outliers are extreme values that differ significantly from the rest of the data. They can have a substantial effect on statistical measures such as the mean and standard deviation, which are used to calculate confidence intervals. If an outlier is present, it can either artificially inflate or deflate the confidence interval, making it unreliable. In small samples, like the one in this study (only 24 rats), a single outlier can skew results dramatically. - Why Other Options Are Less Concerning?
- Mild Right Skew in a Stemplot: A slight skew does not necessarily invalidate the confidence interval, especially if the sample size is moderate (n=24). The CLT states that for sufficiently large samples, the sampling distribution of the mean will approximate normality even if the original population distribution is slightly skewed.
- Unknown Population Standard Deviation: When the population standard deviation is unknown, a t-distribution is used instead of a normal distribution. This is standard practice and does not invalidate the interval.
- Population Distribution Not Exactly Normal: The normality assumption is less critical when the sample size is moderate or large due to the CLT.
Conclusion:
Among the given choices, an outlier is the biggest threat to the validity of the confidence interval. It can distort both the mean and standard deviation, leading to misleading results. Therefore, careful examination of data for outliers is crucial before constructing confidence intervals.