Some antique violins are worth millions of dollars They are prized by music lovers for their uniquely rich, full sound. An audience of experts took part in a blind test of violins, one of which was an expensive antique. There were four other violins (modern-day instruments) made of specially treated wood. When asked to pick the antique violin after listening to all five violins, 41 got it right and 116 got it wrong. Complete parts a and b.
a. If this group were just guessing, how many people (out of the 157) would be expected to guess correctly? And how many would be expected to guess incorrectly?

The expected number of correct guesses is 314
(Type an integer or a decimal.)
The expected number of incorrect guesses is 125.6
(Type an integer or a decimal.)
b. Calculate the observed value of the chi-square statistic for these violins
$\mathrm{x}^{2}=\square($ Round to two decimal places as needed.)

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 \(\boxed{3.67}\).