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## 3.4 Confidence Intervals for the Population Mean

As stressed before, we will never estimate the *exact* value of the population mean of \(Y\) using a random sample. However, we can compute confidence intervals for the population mean. In general, a confidence interval for an unknown parameter is a recipe that, in repeated samples, yields intervals that contain the true parameter with a prespecified probability, the *confidence level*. Confidence intervals are computed using the information available in the sample. Since this information is the result of a random process, confidence intervals are random variables themselves.

Key Concept 3.7 shows how to compute confidence intervals for the unknown population mean \(E(Y)\).

### Key Concept 3.7

### Confidence Intervals for the Population Mean

A \(95\%\) confidence interval for \(\mu_Y\) is a random variable that contains the true \(\mu_Y\) in \(95\%\) of all possible random samples. When \(n\) is large we can use the normal approximation. Then, \(99\%\), \(95\%\), \(90\%\) confidence intervals are

\[\begin{align} &99\%\text{ confidence interval for } \mu_Y = \left[ \overline{Y} \pm 2.58 \times SE(\overline{Y}) \right], \\ &95\%\text{ confidence interval for } \mu_Y = \left[\overline{Y} \pm 1.96 \times SE(\overline{Y}) \right], \\ &90\%\text{ confidence interval for } \mu_Y = \left[ \overline{Y} \pm 1.64 \times SE(\overline{Y}) \right]. \end{align}\]

These confidence intervals are sets of null hypotheses we cannot reject in a two-sided hypothesis test at the given level of confidence.

Now consider the following statements.

In repeated sampling, the interval \[ \left[ \overline{Y} \pm 1.96 \times SE(\overline{Y}) \right] \] covers the true value of \(\mu_Y\) with a probability of \(95\%\).

We have computed \(\overline{Y} = 5.1\) and \(SE(\overline{Y})=2.5\) so the interval \[ \left[ 5.1 \pm 1.96 \times 2.5 \right] = \left[0.2,10\right] \] covers the true value of \(\mu_Y\) with a probability of \(95\%\).

While 1. is right (this is in line with the definition above), 2. is wrong and none of your lecturers wants to read such a sentence in a term paper, written exam or similar, believe us.
The difference is that, while 1. is the definition of a random variable, 2. is one possible *outcome* of this random variable so there is no meaning in making any probabilistic statement about it. Either the computed interval does cover \(\mu_Y\) *or* it does not!

In `R`, testing of hypotheses about the mean of a population on the basis of a random sample is very easy due to functions like `t.test()` from the `stats` package. It produces an object of type `list`. Luckily, one of the most simple ways to use `t.test()` is when you want to obtain a \(95\%\) confidence interval for some population mean. We start by generating some random data and calling `t.test()` in conjunction with `ls()` to obtain a breakdown of the output components.

```
# set seed
set.seed(1)
# generate some sample data
<- rnorm(100, 10, 10)
sampledata
# check the type of the outcome produced by t.test
typeof(t.test(sampledata))
#> [1] "list"
# display the list elements produced by t.test
ls(t.test(sampledata))
#> [1] "alternative" "conf.int" "data.name" "estimate" "method" "null.value" "p.value" "parameter"
#> [9] "statistic" "stderr"
```

Though we find that many items are reported, at the moment we are only interested in computing a \(95\%\) confidence set for the mean.

```
t.test(sampledata)$"conf.int"
#> [1] 9.306651 12.871096
#> attr(,"conf.level")
#> [1] 0.95
```

This tells us that the \(95\%\) confidence interval is

\[ \left[9.31, 12.87\right]. \]

In this example, the computed interval obviously does cover the true \(\mu_Y\) which we know to be \(10\).

Let us have a look at the whole standard output produced by `t.test()`.

```
t.test(sampledata)
#>
#> One Sample t-test
#>
#> data: sampledata
#> t = 12.346, df = 99, p-value < 2.2e-16
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> 9.306651 12.871096
#> sample estimates:
#> mean of x
#> 11.08887
```

We see that `t.test()` does not only compute a \(95\%\) confidence interval but automatically conducts a two-sided significance test of the hypothesis \(H_0: \mu_Y = 0\) at the level of \(5\%\) and reports relevant parameters thereof: the alternative hypothesis, the estimated mean, the resulting \(t\)-statistic, the degrees of freedom of the underlying \(t\) distribution (`t.test()` does use perform the normal approximation) and the corresponding \(p\)-value. This is very convenient!

In this example, we come to the conclusion that the population mean *is* significantly different from \(0\) (which is correct) at the level of \(5\%\), since \(\mu_Y = 0\) is not an element of the \(95\%\) confidence interval

\[ 0 \not\in \left[9.31,12.87\right]. \] We come to an equivalent result when using the \(p\)-value rejection rule since

\[ p\text{-value} = 2.2\cdot 10^{-16} \ll 0.05. \]