Once a brain-to-brain connection was established, a pair of rats would undergo 50 lever-pressing trials. At each trial, the researchers would record whether or not the encoder rat pressed the "correct" lever and whether or not the decoder rat also pressed the "correct" lever. Following is a summary of the data.

(a) Consider the question: Is decoder accuracy related to encoder accuracy? That is, does the decoder accuracy depend on whether or not the encoder pressed the right lever? Express the null hypothesis related to this research question symbolically. You may use the notation EC, EW, DC, and DW as needed to describe the groups (e.g., EC = Encoder correct).
(b) Complete the table of expected counts used for the χ2 test of independence.
(c) The value of χ2s = 1.801 for these data. Is there statistically significant evidence (α = 0.05) that the decoder is more likely to press the correct lever if the encoder also presses the correct lever than if the encoder does not? Justify your answer.
(d) There is one condition for the χ2-test of independence that is clearly violated. Which condition is violated?

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Decoder Decoder Total 38 12 50 correct wrong 27 Encoder correct Encoder wrong Total 17