Final answer:
Based on whether the confidence intervals overlap, we can infer if the samples might have the same population means. A lack of overlap suggests different means, while overlap does not provide conclusive evidence. Ninety percent of confidence intervals should contain the population mean; a 90% CI doesn't mean it contains 90% of the data.
Step-by-step explanation:
Analysis of Confidence Intervals
First, we need to understand what a confidence interval (CI) is. A confidence interval is a range around a sample mean that indicates where the true population mean (μ) is likely to be found. If we say we have a 90 percent confidence interval, it means that if we were to take many samples and build a confidence interval from each of them, about 90 percent of these intervals would contain the true population mean.
Answering whether three confidence intervals suggest that the samples come from populations with the same mean depends on whether the intervals overlap. If the confidence intervals for the sample means have no overlap, it strongly suggests that the population means are different. When intervals do overlap, there’s a possibility that the population means could be equal, but it's not conclusive without further statistical testing.
In the given study with three sample means being compared, if the confidence intervals for each sample mean are entirely distinct with no overlap, it would suggest the samples have been taken from populations with different means. However, if there is overlap between the confidence intervals, we cannot definitely conclude that the means are the same or different.
When looking at the percentage of confidence intervals that contain the true population mean, it is expected to be close to the confidence level, in this case, 90 percent. This means that out of 100 confidence intervals calculated at a 90 percent confidence level, we would anticipate that around 90 of those would contain the true population mean.
It is crucial to note that a common misconception is that a 90 percent CI would contain 90 percent of the data. This is incorrect. The confidence level refers to the percentage of confidence intervals that would contain the true population mean, not the percentage of data points within any single interval.