Problem 1. Consider dealing a hand of 6 cards from a well shuffled standard deck | Solution

Problem 1. Consider dealing a hand of 6 cards from a well shuffled standard deck of playing cards. Let X1 be the number of fours in the hand; let X2 be the number of Queens in the hand; and let X3 be the number of cards in the hand that are neither fours nor Queens. The random variables {X1, X2, X3} have a multinomial joint distribution. (Q1) (True or False)

Problem 2. Consider drawing 7 cards at random with replacment 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)

Problem 3. A bowl contains 19 balls of various colors:
color mauve magenta yellow cyan coral
balls in bowl 1 3 4 7 4
Consider drawing a random sample of size 10 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.
Fill in the expected values
color mauve magenta yellow cyan coral
expected number (Q3) (Q4) (Q5) (Q6) (Q7)
Consider the following possible outcome of drawing the random sample:
color mauve magenta yellow cyan coral
balls in sample 0 1 3 4 2
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 10 with replacement from this bowl? (Q10) (yes or no)
 

Problem 4. The area under the chi-square curve with 48 degrees of freedom from 38.2 to 77.4 is (Q11)
The 70% percentile of the chi-square curve with 48 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:
girth range <3 ft. 3-6 ft. 6-9 ft. 9-12ft >12 ft.
claimed fraction of trees 10% 20% 35% 30% 5%
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 20 locations on the property at random from a map, and send a surveyor to those 20 locations to measure all the trees within a 25 foot radius of the location.
Suppose that this results in measurementsof the girth of 360 trees, and that for all practical purposes, those 360 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 360 trees in the sample is as follows:
girth range <3 ft. 3-6 ft. 6-9 ft. 9-12ft >12 ft.
observed number of trees 30 75 146 96 13
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) (is, is not) 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 10%? (Q18) (yes, no)