A card-dealing machine deals spades (1), hearts (2), clubs (3), and diamonds (4) at random as if from an infinite deck. In a randomness check, 1,600 cards were dealt and counted. The results are shown below. At the 5% significance level, the critical value is ________.

Question

A card-dealing machine deals spades (1), hearts (2), clubs (3), and diamonds (4) at random as if from an infinite deck. In a randomness check, 1,600 cards were dealt and counted. The results are shown below.  At the 5% significance level, the critical value is ________.

Quizplus · Accepted Answer

This question requires us to perform a chi-square goodness of fit test to check whether the observed frequencies of card suits match the expected frequencies. The expected frequencies can be calculated by assuming that the probability of each suit is 0.25 (since there are 4 suits) and multiplying by the total sample size of 1600.
Expected frequency = 0.25 * 1600 = 400
The chi-square statistic for this test is calculated using the formula:
χ2 = Σ((Observed frequency - Expected frequency)^2 / Expected frequency)
We can calculate this statistic for each suit and then add them up to get the total chi-square value. Using the given data, we get:
χ2 = ((365-400)^2/400) + ((411-400)^2/400) + ((437-400)^2/400) + ((387-400)^2/400)
χ2 = 8.61
The critical value for a chi-square test with 3 degrees of freedom (4 suits - 1 = 3) at 5% significance level is 7.815. Since 8.61 is greater than 7.815, we can reject the null hypothesis that the observed frequencies match the expected frequencies and conclude that there is a significant difference between the observed and expected frequencies of card suits. Therefore, the best choice is B: 7.815.

A Card-Dealing Machine Deals Spades (1), Hearts (2), Clubs (3)