This problem examines a hypothetical prospective study about an important problem in the developing world: the use of mosquito nets to prevent malaria in children. The nets are typically used to protect children from mosquitoes while sleeping.

Suppose that in a large region of an African country, 100 households with one child are randomized to receive free mosquito nets for the child in the household and 100 households with one child are randomized to a control group where families do not receive the nets.

You are given the following information:

- In the 100 households receiving the nets, 22 children became infected with malaria.

- In the 100 households without the nets, 30 children became infected with malaria.

- The 200 families selected to participate in the study may be regarded as a random sample from the families in the region, so the 100 families in each group may be regarded as random samples from the population.

- Malaria among children is common in this region, with a prevalence of approximately \(25 \%\).

(a) Write down the \(2 \times 2\) contingency table that corresponds to the data from the trial, labeling the table clearly and including the row and column totals.

(b) Under the hypothesis of no association between use of a mosquito net and malaria infection, calculate the expected number of infected children among 100 families who did receive a net.

(c) The \(\chi^{2}\) statistic for this \(2 \times 2\) table is 1.66 . Use this information to conduct a test of the null hypothesis of no effect of the use of a mosquito net on malaria infection in children.

(d) Compute and interpret the estimated relative risk of malaria infection, comparing the households without a net to those with a net.