Question

On a certain flight, from prior data, 70% of passengers will buy a meal. A typical row in the economy section of this flight seats 10 in "3-4-3" seating. Create an Excel spreadsheet. In your Excel spreadsheet, answer the following questions: a. What method was used to determine that 70% of passengers will buy a meal? b. Use the letter M to stand for your random variable. What is the meaning of M? The answer to this question begins, "Let M = ...." c. What probability distribution is appropriate to assign to M? d. Compute the probability mass function, pmf, of that probability distribution from (c) above using an Excel formula. The values of the pmf should be in one vertical column (see the attached PDF for an example). NOTE 1: You can put n and p somewhere on your Excel sheet. Then, in your computation of the pmf, using =BINOM.DIST(), you should reference (and "lock") the cells. NOTE 2: Looking below, you can see that you are asked to create the cdf adjacent to this pmf. If you lock the column here in (d), then you can copy the =BINOM.DIST() formula to the adjacent column, and simply change the FALSE to TRUE. Then, you can propagate the formula down and be done. e. Compute the cumulative distribution function, cdf, using an Excel formula. The values of the cdf should be in one vertical column, adjacent to the pdf. f. Verify Excel's formula in d. as follows: Create one column based on Excel's version of nCr = COMBIN(n, r) Create an adjacent column that computes the probability of the successes Create an adjacent column that computes the probability of the failures Multiply the three factors and verify that this matches the numerical value in d g. In a typical row in economy, how many passengers would you expect to purchase a meal? h. How likely is this to occur? i. Compute the interval one standard deviation, σ, above and one standard deviation below this expected value. j. What is the probability of the interval P(μ - σ ≤ M ≤ μ + σ) = ? k. What is the probability that at least one passenger buys a meal in a typical row in economy? l. What is the probability that no passenger buys a meal in a typical row in economy? Upload your Excel workbook file to this assignment. NOTE: Two OPTIONAL opportunities for extra credit: I. Create a graph of the probability mass function, pmf, the "comb" function. II. Create a graph of the cumulative probability distribution, cdf, the step function (HARDER). You may do either one or both for extra credit.

Answer 1

SOLUTION

A) Random Sampling

B) Let M = Number of passengers in a randomly selected row of the economy section of the given flight who will buy a meal

C) Binomial Distribution (Bernoulli trials)

D) E) With n=10 and p=0.7, the PDF and CDF are obtained as

PDF = fM(m)="C,m p"(1 -p"m. 1-m n 10, p0.7, 0<m <10

CDF F(m) "C, p(1-p)

n	p	m	PDF	CDF
10	0.7	0	5.9049E-06	5.9049E-06
		1	0.000137781	0.000143686
		2	0.001446701	0.001590386
		3	0.009001692	0.010592078
		4	0.036756909	0.047348987
		5	0.102919345	0.150268333
		6	0.200120949	0.350389282
		7	0.266827932	0.617217214
		8	0.233474441	0.850691654
		9	0.121060821	0.971752475
		10	0.028247525	1

-BINOM.DIST(C2,$A$2,$B$2, FALSE) D2 A B C D 1 PDF CDF nCr 0.7 2 10 0 5.9049E-06 5.9049E-06 1 10 3 1 0.000137781 0.000143686 45 4 2 0.001446701 0.001590386 5 0.009001692 0.010592078 120 6 4 0.036756909 0.047348987 210 7 5 0.102919345 0.150268333 252 6 0.200120949 0.350389282 210 7 0.266827932 0.617217214 120 10 8 0.233474441 0.850691654 45 10 11 9 0.121060821 0.971752475 12 10 0.028247525 1 1 et LC LO

f. The mentioned computation of probability, using product of all terms is obtained as

Probability "Cm X p" x (1- p)" n-m 0.7 = 10, p

x	PDF	CDF	nCx	p^x	(1-p)^(n-x)	Probability
0	5.9049E-06	5.9049E-06	1	1	5.9049E-06	5.9049E-06
1	0.000137781	0.000143686	10	0.7	1.9683E-05	0.000137781
2	0.001446701	0.001590386	45	0.49	6.561E-05	0.001446701
3	0.009001692	0.010592078	120	0.343	0.0002187	0.009001692
4	0.036756909	0.047348987	210	0.2401	0.000729	0.036756909
5	0.102919345	0.150268333	252	0.16807	0.00243	0.102919345
6	0.200120949	0.350389282	210	0.117649	0.0081	0.200120949
7	0.266827932	0.617217214	120	0.082354	0.027	0.266827932
8	0.233474441	0.850691654	45	0.057648	0.09	0.233474441
9	0.121060821	0.971752475	10	0.040354	0.3	0.121060821
10	0.028247525	1	1	0.028248	1	0.028247525

f F2*G2*H2 12 A C D E G H (1-p)^(n-m) Probability 1 PDF CDF nCm pAm p 2 10 0.7 0 5.9049E-06 5.9049E-06 1 1 5.9049E-06 5.9049E-06 0.000143686 10 1 0.000137781 0.7 0.000019683 0.000137781 4 0.001446701 2 0.001590386 45 0.49 6.561E-05 0,001446701 0.009001692 0.009001692 5 3 0.010592078 120 0.343 0.0002187 0.036756909 6 4 0.047348987 210 0.2401 0.000729 0.036756909 0.16807 0.102919345 7 5 0.102919345 0.150268333 252 0.00243 0.200120949 6 0.200120949 0.350389282 210 0.117649 0.0081 7 0.027 0.266827932 0.617217214 120 0.0823543 0.266827932 0.233474441 10 8 0.233474441 0.850691654 45 0.05764801 0.09 10 11 9 0.121060821 0.971752475 0.040353607 0.3 0.121060821 12 10 0.028247525 1 1 0.028247525 1 0.028247525 st

The two columns match exactly, as desired.

g. This is the expected value of Binomial distribution

E(M) np = 10 x 0.7 7

h. This is the probability that 7 passengers will buy a meal. From the table above,

PM 7) 0.266828

i. First we need to compute the standard deviation of the Binomial distribution

= Vnp(1 - p) = V10 x 0.7 x 0.3 1.45

Since this is a discrete distribution, hence we must count to the next integer. Hence, the interval should span 2 points around the mean. Since the mean is 7, hence the interval is from 5 to 9, inclusive.

j. The required probability is computed using the difference of CDF across the range.

M7+2) (u - o< M <u0)- P(7- 2

FM(9) FM(4)

0.9244

k. This is computed as the complementary probability that no passenger buys a meal

PM1)1 P(M 0) 0.9999941

l. This is simply the probability that M=0

PM 0)0.0000059

PDF 0.30 0.25 0.20 0.15 PDF 0.10 0.05 0.00 C 1 2 4. 5 6 7 8 10

CDF 1 0.9 0.8 0.7 0.6 0.5 CDF 0.4 0.3 0.2 0.1 0 0 1 2 4. 5 6 7 8 10 LC