MATLAB: An Introduction with Applications
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Author: Amos Gilat
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Transcribed Image Text:### Understanding Confidence Intervals
#### Explanation
Determine whether the statement below makes sense or does not make sense. Explain clearly.
**Statement:**
Based on our sample, the 95% confidence interval for the mean amount of television watched by adults in a nation is 2.9 to 4.3 hours per day. Therefore, there is a 95% chance that the mean for all adults in the nation will fall somewhere in this range and a 5% chance that it will not.
---
#### Multiple Choice Question
Choose the correct answer below.
**A.** The statement makes sense. There is a 5% probability that the confidence interval limits do not contain the true value of the sample mean, so the probability it does not contain the true value of the population mean is also 5%.
**B.** The statement does not make sense. The population mean is a fixed constant that either falls within the confidence interval or it does not. There is no probability associated with this.
**C.** The statement does not make sense. The probability the population mean is greater than the upper limit is 5% and the probability it is less than the lower limit is 5%, so the probability it does not is 5% + 5% = 10%.
**D.** The statement makes sense. There is 95% probability that the confidence interval limits actually contain the true value of the population mean, so the probability it does not fall in this range is 100% − 95% = 5%.
---
#### Analysis
A confidence interval, in this context, gives a range derived from sample data within which we are *95% certain* that the true population mean lies. This does not imply a probabilistic measure as applicable to the truth of mean if viewed as fixed but rather on the method’s efficacy over repeated sampling.
**Correct Answer:**
**B.** The statement does not make sense. The population mean is a fixed constant that either falls within the confidence interval or it does not. There is no probability associated with this.
Understanding confidence intervals is crucial in statistics, signifying through numerous samples, 95% will contain the true population mean, underpinning methodological reliability without assigning probability to specific interval outcomes.
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