Martijn Wieling

University of Groningen

- How to reason about the population using a sample (CLT)
- Calculating the standard error (\(SE\))
- Standard error: used when reasoning about the population using a
**sample** - Standard deviation: used when comparing an
**individual**to the population

- Standard error: used when reasoning about the population using a
- Calculating a confidence interval
- Specifying a concrete testable hypothesis based on a research question
- Specifying the null (\(H_0\)) and alternative hypothesis (\(H_a\))
- Conducting a \(z\)-test and using the results to evaluate a hypothesis
- Definition of a \(p\)-value: probability of data given that \(H_0\) is true
- Evaluating the statistical significance given \(p\)-value and \(\alpha\)-level
- Difference between a one-tailed and a two-tailed test
- Type I and II errors

- Introduction to \(t\)-test
- Three types of \(t\)-tests:
- Single sample \(t\)-test
- Independent samples \(t\)-test
- Paired samples \(t\)-test

- Effect size
- How to report?

- Last lecture: \(z\)-test is used for comparing averages when \(\sigma\) is known
- \(\sigma\) is only known for standardized tests, such as IQ tests

- When \(\sigma\) is not known (in most cases), we can use the \(t\)-test
- This test includes an estimation of \(\sigma\) based on sample standard deviation \(s\)

- Very similar to calculating \(z\)-value for a sample (using standard error):

\[t = \frac{m - \mu}{s / \sqrt{n}} \hspace{70pt} z = \frac{m - \mu}{\sigma / \sqrt{n}}\]

- Only difference: sample standard deviation \(s\) is used instead of \(\sigma\)
- The precise formula depends on the type of \(t\)-test (independent samples, etc.)
- (But for the exam, you only have to know the basic formulas shown above)

- \(z\)-values are compared to the standard normal distribution
- But \(t\)-values are compared to the \(t\)-distribution
- \(t\)-distributions look similar to the standard normal distribution
- but dependent on the number of
**degrees of freedom (dF)**

- but dependent on the number of

- There are five balloons each having a different color
- There are five students (\(n = 5\)) who need to select a balloon
- If 4 students have selected a balloon (
**dF = 4**), student nr. 5 gets the last balloon

- If 4 students have selected a balloon (
- Similarly: if we have a
*fixed*mean value calculated from 10 values- 9 values may vary in their value, but the 10th is fixed: dF = 10 - 1 = 9

- Difference between normal distribution and \(t\)-distribution is large for small dFs
- When dF \(\geq\) 100, the difference is negligible
- As the shape differs, the \(p\)-value associated with a certain \(t\)-value also changes
- That is why it is
**essential**to specify dF when describing the results of a \(t\)-test:

\(t\)(dF)

- For significance (given \(\alpha\)), higher (abs.) \(t\)-values are needed than \(z\)-values (but only when dF < 100, otherwise \(z\) and \(t\) are equal)

```
qt(0.025, df = 10, lower.tail = F) # crit. t-value (alpha = 0.025) for dF = 10
```

```
# [1] 2.2281
```

```
pt(2, 10, lower.tail = F) * 2 # two-sided p-value = 2 * one-sided p-value
```

```
# [1] 0.073388
```

- Dark gray area: \(p\) < 0.05 (2-tailed)

*Single sample \(t\)-test*: compare mean with fixed value*Independent sample \(t\)-test*: compare the means of two**independent**groups*Paired \(t\)-test*: compare pairs of (dependent) values (e.g., repeated measurements of same subjects)**Requirement for all \(t\)-tests**: Data should be approximately normally distributed- Otherwise: use non-parametric tests (discussed in next lecture)

\[t = \frac{m - \mu}{s / \sqrt{n}}\]

- Used to compare mean to fixed value
- \(H_0\): \(\mu = \mu_0\) and \(H_a\): \(\mu \neq \mu_0\)
- Larger \(t\)-values give reason to reject to \(H_0\)
- Automatic calculation in
`R`

using function`t.test()`

- Standardized effect size is measured as difference in standard deviations
- Cohen's \(d\): \(d = (m - \mu) / s\)

- Data randomly selected from population
- Data measured at interval or ratio scale
- Observations are independent
- Observations are approximately normally distributed
- But \(t\)-test is robust to non-normality for larger samples (\(n > 30\))

- Given our English proficiency data, we'd like to assess if the average English score is different from 7.5
- \(H_0\): \(\mu = 7.5\) and \(H_a\): \(\mu \neq 7.5\)
- We use \(\alpha\) = 0.05
- Sample mean \(m\) = 7.62
- Sample standard deviaton \(s\) = 0.92
- Sample size \(n\) = 500
- Degrees of freedom of \(t\)-test equals 500 - 1 = 499

- Data randomly selected from population ?
- Data measured at interval scale ✓
- Independent observations ✓
- Data roughly normally distributed (or > 30 observations) ✓