# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 20

Since $\, \mu \,$ is a fixed number it does not make sense that $\, P(1.44 < \mu < 1.76) = 0.95 \,$. Just to make it clear - probability means that there is a range/set of values that a random variable can take. Here the population mean is a fixed value, thus it does not make sense to say $\, P(1.44 < \mu < 1.76) = 0.95 \,$.

The confidence interval $\, (1.44, 1.76) \,$ means that 95% of the confidence intervals formed in this way contains $\, \mu \,$. Or there is a 95% chance that this interval contains $\, \mu \,$ but it does not say that the probability that $\, \mu \,$ takes one of the values in the interval $\, (1.44, 1,76) \,$ is $\, 0.95 \,$. The value of $\, \mu \,$ is fixed and it does not take any values from the interval! It’s the interval that we say that the chance that this interval containing the value of $\, \mu \,$ is 95% because intervals are random and the interval we formed in this way have the probability for containing $\, \mu \,$ is 95%.

The confidence interval says about the probability of the interval not of the population mean.

$$\tag*{\blacksquare}$$