5.01 Quiz: Using Chi-Square for Goodness-of-Fit

Choose the answer. Which of the following are true of the components of the chi-square statistic (X^2)? 1. ∑(O - E) = 0 2. ∑(O - E)2 = X² 3. ∑(O - E)2 = 0 when data matches expected values.

1 and 3 only

You want to know whether birthdays are equally distributed through all 12 months, so you take a random sample of 149 people and record each person's birth month. What would you use as the expected value for a chi-square goodness-of-fit test of these data?

12.417

You want to know if birthdays are equally distributed through all 12 months. You take a random sample of 149 people and record the following number of birthdays in each month: Jan: 14 Feb: 15 Mar: 13 Apr: 13 May: 14 Jun: 7 Jul: 5 Aug: 17 Sep: 13 Oct: 9 Nov: 11 Dec: 18 You calculate an expected value of 12.417 people for each month. Calculate X2 for the goodness-of-fit test for the following hypotheses: H0: Each month has a proportion of .083 birthdays. Ha: At least one month does not have a proportion of .083 birthdays.

13.12

The spinner for a board game has eight colors the arrow can land on. What's the minimum number of spins you'd need to be able to perform a chi-square goodness-of-fit test?

40

Your friend says she has an unfair number cube: the probability of getting a one or a six is 1/3 for each, and the probability of getting a two, three, four, or five is 1/12 for each. You want to test her statement. What's the minimum number of times you have to roll the number cube to use a chi-square goodness-of-fit test here?

60

The spinner for a board game has eight colors the arrow can land on. What are the degrees of freedom for a goodness-of-fit test of the fairness of this spinner?

7

Which of the following is not true of chi-square distributions and the goodness-of-fit test?

Chi-square distributions approximate normal distributions.

Your friend says she has an unfair number cube: the probability of getting a one or a six is 1/3 for each, and the probability of getting a two, three, four, or five is 1/12 for each. You want to test her statement. If you roll the number cube 96 times, what are the expected values for each number on the number cube?

The expected values are 32 for both ones and sixes, and 8 for twos, threes, fours and fives.

Your friend says she has an unfair number cube: the probability of getting a one or a six is 1/3 for each, and the probability of getting a two, three, four, or five is 1/12 for each. You want to test her statement, so you roll the number cube 96 times and get the following results: 1 : 26 2 : 6 3 : 7 4 : 13 5 : 10 6 : 34 The expected values are 32 for both ones and sixes, and 8 for twos, threes, fours, and fives. Calculate your chi-square statistic and draw a conclusion for a test of the following hypotheses with alpha equals.1: H subscript 0 : The probability of getting a one or a six is 1 third for each, and the probability of getting a two, three, four or five is 1 over 12 for each. H subscript a : At least one of the hypothesized probabilities is incorrect.

X squared equals 5.5. We can't reject the null hypothesis that the probability of getting a one or six is 1 third and that the probability of getting a two, three, four or five is 1 over 12. With six categories, a X squared of 5.5 or greater will occur due to chance alone more than 25% of the time

The spinner in a board game has eight colors the arrow can land on. To test the fairness of the spinner, you spin the arrow 75 times and get the following results: Green: 3 Blue: 13 Red: 9 Orange: 9 Brown: 16 Yellow: 14 White: 8 Black: 3 Calculate the chi-square statistic for these data and use a table to find the P-value.

X² = 17.267, .02 > p > .01