There is a 95% chance that \(\mu\) is between 10.2 and 15.6 seconds
The population mean reaction time \(\mu\) will fall within [10.2, 15.6] seconds 95% of the time.
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False. The population mean is a FIXED but unknown number. It’s unknown because we don’t have the entire population data. But if we had the data, we could easily compute the mean and find that number - which won’t change if you re-compute it.
True. The sample mean is a random variable and has a probability distribution. We can compute probabilities for the sample mean.
No, that interpretation is wrong. The population mean \(\mu\) is not a random variable, it’s a fixed number (even if unknown). Probability statements can only be written when referred to random variables. The statement presented in Question 3 involves \(\mu\) (a fixed but unknown number) and two computed numbers (10.2 and 15.6). There is no random variable involved, hence we cannot speak of probability. Once we compute a confidence interval, the population mean \(\mu\) either does or does not fall within it, we simply won’t know for sure.
The correct interpretation would be:
We are 95% confident that the population mean reaction time is between 10.2 and 15.6 seconds.
No, that interpretation is wrong. The correct interpretation would be:
We are 95% confident that the population mean reaction time is between 10.2 and 15.6 seconds.