Step 1 :The problem is asking for two things. First, it wants to know the expected number of correct and incorrect guesses if the group were just guessing. Since there are 5 violins and the group is guessing, the probability of guessing correctly is 1/5 and the probability of guessing incorrectly is 4/5. We can calculate the expected number of correct and incorrect guesses by multiplying these probabilities by the total number of guesses, which is 157.
Step 2 :Second, it wants to calculate the observed value of the chi-square statistic for these violins. The chi-square statistic is a measure of how much observed data deviates from expected data. It is calculated by summing the squares of the differences between observed and expected data, divided by the expected data, for each category. In this case, the categories are 'correct guess' and 'incorrect guess'.
Step 3 :Let's calculate the expected number of correct and incorrect guesses. The total number of guesses is 157. The probability of guessing correctly is 1/5, so the expected number of correct guesses is \(157 \times \frac{1}{5} = 31.4\). The probability of guessing incorrectly is 4/5, so the expected number of incorrect guesses is \(157 \times \frac{4}{5} = 125.6\).
Step 4 :Now, let's calculate the observed value of the chi-square statistic. The observed number of correct guesses is 41 and the observed number of incorrect guesses is 116. The chi-square statistic is calculated as \(\chi^{2} = \sum \frac{(O-E)^{2}}{E}\), where O is the observed data and E is the expected data. For the correct guesses, this is \(\frac{(41-31.4)^{2}}{31.4} = 2.93\). For the incorrect guesses, this is \(\frac{(116-125.6)^{2}}{125.6} = 0.74\). Summing these gives a chi-square statistic of \(2.93 + 0.74 = 3.67\).
Step 5 :Final Answer: The expected number of correct guesses is \(\boxed{31.4}\) and the expected number of incorrect guesses is \(\boxed{125.6}\). The observed value of the chi-square statistic for these violins is \(\boxed{3.67}\).