Question

(6 points) The following percentages were obtained over many years of observation by the U.S. Weather Bureau. All data listed are for the month of December. Location Long-Term Mean % of Clear Days in Dec. Juneau, Alaska 18% Seattle, Washington 24% Hilo, Hawaii 36% Las Vegas, Nevada 75% Phoenix, Arizona 77% In the locations listed, the month of December is a relatively stable month with respect to weather. Since weather patterns in these locations from one day to the next are more or less the same, it is reasonable to use a binomial probability model. Let r be the number of clear days in December. Since December has 31 days, 0≤r≤31. P(r) is the binomial probability for each of the listed locations when r=0,1,2,… ,31. Include your formula setup or calculator comments to answer the following questions. (1 pt) Find the probability that Juneau will have at most 8 clear days in December. (1 pt) Find the probability the Hilo will have at least 14 clear days in December. (1 pt) Find the probability that Phoenix will have 20 or more clear days in December. (1 pt) Find the probability that Las Vegas will have no more than 18 clear days in December. (2 pts) Find the probability that Seattle will have from 5 to 10 (including 5 and 10) clear days in December

Answer 1

solution:

The cumulative probability distribution function for binomial is defined as,

$P(X\leq k)=\sum_{k=0}^{k} \binom{n}{k}p^k(1-p)^{n-k}$

a)

For juneau,

probability of success, p = 0.18

For at most 8 clear days in December, k<=8

$P(X\leq 8)=\sum_{k=0}^{8} \binom{31}{k} 0.18^k(1-0.18)^{31-k}$

The calculation is excel using the function, = BINOM.DIST(8,31,0.18 TRUE). The screenshot is shown below,

BQ2 X f r =BINOM.DIST(8,31,0.18, TR BS BT BO BQ BR BP Juneau P(X<=8) 0.908973

$P(X\leq 8)=0.9089$

b) For Hilo,

Probability of success, P=0.36

For at least 14 clear days in December, k>=14

$P(X\geq 14)=1-P(X<= 13)=1-\sum_{k=0}^{13} \binom{31}{k} 0.36^k(1-0.36)^{31-k}$

The calculation is done in excel using the function , = 1-BINOM.DIST(13,31,0.36,TRUE).

$P(X\geq 14)=0.1896$

c)

For phonenix,

Probability of success, p=0.77

For 20 more clear days in December, k>=20

$P(X\geq 20)=1-P(X<= 19)=1-\sum_{k=0}^{19} \binom{31}{k} 0.77^k(1-0.77)^{31-k}$

The calculation is done in excel using the function =1-BINOM.DIST(19,31,0.77,TRUE).

$P(X\geq 14)=0.9637$

d)

For Las Vegas,

Probability of success, P=0.75

For no more than 18 clear days in December,K<=18

$P(X\leq 18)=\sum_{k=0}^{18} \binom{31}{k} 0.75^k(1-0.75)^{31-k}$

The calculation is done in excel using the function , = BINOM.DIST(18,31,0.75,TRUE).

$P(X\leq 18)=0.02886$

e)

For seattle,

Probability of success, P= 0.24

For clear days between 5 to 10 in December, 5<=k<=10

$P(5\geqX\leq10)=P(X\leq10)-P(X\leq5)$

$P(5\geqX\leq10)=\sum_{k=0}^{10} \binom{31}{k} 0.24^k(1-0.24)^{31-k}-\sum_{k=0}^{5} \binom{31}{k} 0.24^k(1-0.24)^{31-k}$

The calculation is done in excel using the function, = BINOM.DIST(10,31,0.24,TRUE)-

BINOM.DIST(5,31,0.24,TRUE)

$P(X\leq 18)=0.68677$

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