*To be solved at home before the exercise session.*

- Assume that we have an iid. random sample \(x_1, \ldots , x_{1000}\) and we’d like to use the normal Q-Q plot to assess whether the sample came from a normal distibution. How do you expect the normal Q-Q plot to roughly look like (i.e. what general features do you expect it to have and
*why*), if the true distribution of the data is- a normal distribution,
- a right-skew distribution,
- a left-skew distribution,
- a bimodal distribution,
- a distribution with light tails,
- a distribution with heavy tails?

- Recall the differences between the interpretations of the \(\chi^2\) homogeneity test and \(\chi^2\) test for independence. Come up with a practical situation where the collected data can be expressed as a 2-by-2 table and a related research question for which the correct interpretation is through
- the \(\chi^2\) homogeneity test,
- the \(\chi^2\) test for independence.

- Assume that we have an iid. random sample \(x_1, \ldots , x_{1000}\) and we’d like to use the normal Q-Q plot to assess whether the sample came from a normal distibution. How do you expect the normal Q-Q plot to roughly look like (i.e. what general features do you expect it to have and

*To be solved at the exercise session.*

*Note: all the needed data sets are either given below or available in base R.*

- The data set
`rock`

contains measurements on 48 rock samples from a petroleum reservoir. Treat the data as an iid. random sample from some distribution and test whether the distribution of`shape`

is normal.- Visualize the data to obtain a preliminary idea of the possible normality of the data.
- Use the normal Q-Q plot to gain more evidence on the normality/non-normality of the data.
- Conduct the Bowman-Shenton (Jarque-Bera) and the Shapiro-Wilk tests of normality on significance level 0.05.
- After all the previous, would you conclude the data to be normal (or normal enough for methods with normality assumptions)?
- Why is the data not really iid.?

The data set

`randu`

contains 400 triples of successive random numbers from the random number generator RANDU. Use the \(\chi^2\) goodness-of-fit test to assess whether the first elements in the triplets really obey the uniform distribution on \([0, 1]\).- Extract the first elements in the triplets and visualize their sample distribution.
- Discretize the values into a suitable number of categories and calculate the observed category frequencies.
- Compute the corresponding expected category probabilites under the uniform distribution on \([0, 1]\).
- Recall the hypotheses of the test and conduct it on significance level 0.05.
- What are the conclusions of the test? Compare your results with someone who used a different choice of categories for the discretization.

- The data set
`Titanic`

contains information on the fate of passengers on the fatal maiden voyage of the ocean liner “Titanic”. We use the data to study whether there is a connection between the sex (Male/Female) of a passenger and surviving from the ship (No/Yes).- Extract a marginal table containing only the cross-tabulation of the variables
`Sex`

and`Survived`

. - Find a suitable way to visualize the data.
- Which test is appropriate for these data (and why?), \(\chi^2\) homogeneity test or the \(\chi^2\) test for independence?
- Conduct your chosen test on significance level 0.05 and state your conclusions.

- Extract a marginal table containing only the cross-tabulation of the variables

**(Optional)**Choose your favorite non-normal distribution and use simulations to study the Type II error probabilities of the Bowman-Shenton (Jarque-Bera) and Shapiro-Wilk tests of normality for that distribution on different sample sizes (e.g. \(n = 10, 100, 1000, 10000\)). That is, find out the probabilty of falsely concluding that the data comes from a normal distribution when it does not.