Problem 3. A bowl contains 15 balls of various colors: Consider drawing a random

Question

Problem 3. A bowl contains 15 balls of various colors:

Consider drawing a random sample of size 9 with replacement from the bowl.

Fill in the following table of expected values of the numbers of balls of each color in the random sample.

Consider the following possible outcome of drawing the random sample:

The probability of this outcome is (Q7)

The chi-squared statistic for this outcome is (Q8)

Is the chi-square curve with 3 degrees of freedom a good approximation to the probability histogram of the chi-squared statistic for a random sample of size 9 with replacement from this bowl? (Q9) ? A: no B: yes

Explanation / Answer

Expected Number of grey balls = (Number of gray balls)/ (Number of total balls) * Number of chances

Q3 = (3/15) * 9 = 1.8

Q4 = E (charteuse) = (1/15) * 9 = 0.6

Q5 = E(Orange) = (6/15) * 9 = 3.6

Q6 = E(Turquiose) = (5/15) * 9 = 3

Q7 = Probability of the outcome is = 9P1 * (3/15)1 9P1 * (1/15)1 * 9P5 * (6/15)5 * 9P2 * (2/15)2 = 0.68

Chi squared statistic here = (Observed - Expected)2 / Expected = Q8

(1- 1.8)2 /1.8 + (1- 0.6)2/ 0.6+ ( 5 - 3.6)2 /3.6 + (2-3)2 /3 = 1.5

Yes, the chi-square curve with 3 degrees of freedom a good approximation to the probability histogram of the chi-squared statistic for a random sample of size 9 with replacement from this bowl

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