The Multinomial Distribution and the Chi-Squared test for Goodness of Fit

Problem 1. Consider dealing a hand of 3 cards from a well shuffled standard deck of playing cards. Let X₁ be the number of eights in the hand; let X₂ be the number of nines in the hand; and let X₃ be the number of cards in the hand that are neither eights nor nines. The random variables {X₁, X₂, X₃} have a multinomial joint distribution. (Q1)   ? A: false B: true 

Problem 2. Consider drawing 6 cards at random with replacement from a standard deck of playing cards. For this problem, consider Aces to be face cards, not numbered cards. Which of the following sets of variables have a multinomial joint distribution? (select all such sets)

(Q2)   ? A: the number of kings drawn, and the number of other cards drawn B: the number of kings drawn, the number of hearts drawn, and the number of twos drawn C: the number of face cards drawn, the number of even-numbered cards drawn, and number of odd-numbered cards drawn D: the number of spades drawn, the number of hearts drawn, and the number of diamonds drawn E: the number of face cards drawn, the number of kings drawn, and the number of other cards drawn 

Problem 3. A bowl contains 14 balls of various colors:

Consider drawing a random sample of size 6 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 (Q8) 

The chi-squared statistic for this outcome is (Q9) 

Is the chi-square curve with 4 degrees of freedom a good approximation to the probability histogram of the chi-squared statistic for a random sample of size 6 with replacement from this bowl? (Q10)   ? A: no B: yes 

Problem 4. The area under the chi-square curve with 31 degrees of freedom from 23.1 to 38.9 is (Q11) 

The 60% percentile of the chi-square curve with 31 degrees of freedom is (Q12) 

Problem 5. A logging company is bidding for logging rights on a large tract of land. The girth of a tree is the circumference of its trunk at a particular height above the ground (roughly breast height). The girth is important commercially, because it determines the width of the boards that can be milled from the tree. The owner of the property claims that the distribution of tree girth on the property is as follows:

The logging company can estimate the number of trees fairly accurately using aerial photography, but needs to do a survey on the ground to determine the distribution of girth. Ground surveys are expensive, so the logging company would like to draw its conclusions without surveying the entire tract.

The company hires you as an expert statistician to test the hypothesis that the distribution of girth is what the seller claims. You select 25 locations on the property at random from a map, and send a surveyor to those 25 locations to measure all the trees within a 25 foot radius of the location.

Suppose that this results in measurementsof the girth of 240 trees, and that for all practical purposes, those 240 trees can be treated as a random sample with replacement from the population of trees on the property.

The observed distribution of girth among the 240 trees in the sample is as follows:

The observed value of the chi-squared statistic is (Q13) 

Under the null hypothesis, the smallest expected number of trees in any of the categories is (Q14)  so, under the null hypothesis, the probability histogram of the chi-squared statistic (Q15)   ? A: is not B: is  approximated well by a chi-squared curve.

The appropriate chi-squared curve to use to approximate the probability histogram of the chi-squared statistic has (enter a number) (Q16)  degrees of freedom.

The P-value of the null hypothesis is approximately (Q17) 

Should the null hypothesis be rejected at significance level 5%? (Q18)   ? A: no B: yes