### 7. Hypothesis Testing: Practical 7

### One-sample t-test

The most basic statistical test uses observations for one sample to test a population mean against a specific value. The R command to conduct this test is `t.test()`

.

Let's take a look at the documentation`.`

?t.test

The `t.test()`

command can perform one- as well as two-sample t-tests. In this chapter, we will focus on one-sample tests. The most important arguments are of the `t-test()`

function are:

**x**: the vector of data values you want to test,**y**: optional second vector of data values if you perform a two sample test,**alternative**: specify if you want a two-sided, left-tailed or right tailed t-test with respectively the arguments**"two.sided"**,**"less"**and**"greater"**,**mu**: the critical value you want to test your observations against,**paired**: either TRUE or FALSE. The default is FALSE,**conf.level**: confidence level of the interval, the default value is 0.95.

### One sample t-test

Let's look at how the `t.test()`

command works by applying it to the air quality data of Amsterdam.

Test whether the mean **NO2** concentration at the **Amsterdam-Vondelpark** station **exceeds** the legal standard of #40# (μg/m3). Use a one-sample right-tailed t-test on the mean. You can specify the hypotheses as follows

- #H_0#: #\mu \leq 40#
- #H_a#: #\mu > 40#

Use the separate data frames you created in the previous exercises.

`t.test()`

function in R with the following arguments:**x**: You need the dataframe for the NO2 concentration at the Amsterdam-Vondelpark. Don't forget to specify in which column the concentrations are stored.`x = NO2_vp$value`

.**y**: this is a one-sample t-test. So there is no second vector.**alternative**: You want to test if the legal standard is exceeded. This is thus a right tailed t-test.`alternative = "greater"`

.**mu**: the critical value is 40.`mu = 40`

.**paired**: you don't have paired samples; you can keep the default value (=FALSE).**conf.level**: you can keep the default confidence level of 0.95.

t.test(x = NO2_vp$value, alternative = "greater", mu = 40)

This command results in the following output:

```
One Sample t-test
data: NO2_vp$value
```

```
t = -62.374, df = 1817, p-value = 1
alternative hypothesis: true mean is greater than 40
95 percent confidence interval:
23.84871 Inf
sample estimates:
mean of x
24.26389
```

The layout is common to many of the standard statistical tests and a "dissection" is given in the following:

` One Sample t-test`

This should be self-explanatory. It is simply a description of the test that we have asked for. Notice that by omitting the y-argument, t.test() has automatically found out that a one-sample test is desired.

```
data: NO2_vp$value
```

This tells us simply which data are being tested.

```
t = -62.374, df = 1817, p-value = 1
```

This is where it gets interesting. We get the #t# statistic, the associated degrees of freedom, and the exact #p#-value. We can immediately see that #p > 0.05# and thus that (using #\alpha = 0.05#) the null hypothesis cannot be rejected. In other words, we cannot state that the sample mean is significantly greater than #40#.

`alternative hypothesis: true mean is greater than 40`

This sentence contains two important pieces of information: 1) the critical value we wanted to test the mean on (#40#) and 2) that this is a right-sided test ("greater than").

```
95 percent confidence interval:
23.84871 Inf
```

This is the 95% confidence interval for the true mean; that is, the set of (hypothetical) mean values from which the data do not deviate significantly. (The upper boundary says "Inf" (Infinite) because we're running a right-sided test).

```
sample estimates:
mean of x
```

` 24.26389`

Lastly, the estimated mean of the sample is given.

The conclusion we can draw from this one-sample t-test is that **we cannot reject the null hypothesis** and that **the mean NO2 concentration in this sample does thus not exceed the legal standard**.