Complete the following questions in R Markdown and submit as a single .html file (knitted.)
10/52 or 5/26
10/36 or 5/18
Gender and Residence of Customers Males Females Apartment 233 208 Dorm 159 138 With Parent(s) 102 280 Sorority/Fraternity House 220 265 Other 250 146
What is the probability that a customer is male? Write your answer as a fraction or a decimal number rounded to four decimal places.
(233+159+102+220+250)/ (233+159+102+220+250+208+138+280+265+146) = 964/2001
13/5226/5212/52 = 4056/140608 or 3/104
13/51
13/52*25/51 =325/2652 or 25/204
Students in a Basic Math Class Males Females Freshmen 12 12 Sophomores 19 15 Juniors 12 4 Seniors 7 4
What is the probability that a junior female and then a freshmen male are chosen at random? Write your answer as a fraction or a decimal number rounded to four decimal places.
12/97*4/97 = 48/9409
Step 1. What is the probability that a randomly chosen applicant has a graduate degree, given that they are male? Enter your answer as a fraction or a decimal rounded to four decimal places.
52/141
Step 2. If 102 of the applicants have graduate degrees, what is the probability that a randomly chosen applicant is male, given that the applicant has a graduate degree? Enter your answer as a fraction or a decimal rounded to four decimal places. 52/102
A value meal package at Ron’s Subs consists of a drink, a sandwich, and a bag of chips. There are 6 types of drinks to choose from, 5 types of sandwiches, and 3 types of chips. How many different value meal packages are possible? 653=150 packages
A doctor visits her patients during morning rounds. In how many ways can the doctor visit 5 patients during the morning rounds?54321=120 ways
A coordinator will select 5 songs from a list of 8 songs to compose an event's musical entertainment lineup. How many different lineups are possible?87654=6720ways
A person rolls a standard six-sided die 9 times. In how many ways can he get 3 fours, 5 sixes and 1 two?How many ways can Rudy choose 6 pizza toppings from a menu of 14 toppings if each topping can only be chosen once?1413121110*9= 2162160 ways
3 cards are drawn from a standard deck of 52 playing cards. How many different 3-card hands are possible if the drawing is done without replacement?525150=132600 WAYS
You are ordering a new home theater system that consists of a TV, surround sound system, and DVD player. You can choose from 12 different TVs, 9 types of surround sound systems, and 5 types of DVD players. How many different home theater systems can you build?1295= 540
You need to have a password with 5 letters followed by 3 odd digits between 0 - 9 inclusively. If the characters and digits cannot be used more than once, how many choices do you have for your password?262524232254*3 = 473616000
Evaluate the following expression._9 P_4 (9!/(9-4)!)=9!/5!= 987*6=3024
Evaluate the following expression._11 C_8
n!/(r!(n!-r!)=11!(8!(11!-8!)=11!/(8!3!)= (11109)/(32*1)=165
Evaluate the following expression.( _12 P_8)/( _12 C_4 )
The newly elected president needs to decide the remaining 7 spots available in the cabinet he/she is appointing. If there are 13 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?13C7=1716
combn(13,7)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 5 5 5 5 5 5 5 5 5 5 5 5 5 5
## [6,] 6 6 6 6 6 6 6 7 7 7 7 7 7 8
## [7,] 7 8 9 10 11 12 13 8 9 10 11 12 13 9
## [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 5 5 5 5 5 5 5 5 5 5 5 5
## [6,] 8 8 8 8 9 9 9 9 10 10 10 11
## [7,] 10 11 12 13 10 11 12 13 11 12 13 12
## [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 5 5 6 6 6 6 6 6 6 6 6 6
## [6,] 11 12 7 7 7 7 7 7 8 8 8 8
## [7,] 13 13 8 9 10 11 12 13 9 10 11 12
## [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 6 6 6 6 6 6 6 6 6 6 6 7
## [6,] 8 9 9 9 9 10 10 10 11 11 12 8
## [7,] 13 10 11 12 13 11 12 13 12 13 13 9
## [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61] [,62]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 7 7 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 8 9 9 9 9 10 10 10 11
## [7,] 10 11 12 13 10 11 12 13 11 12 13 12
## [,63] [,64] [,65] [,66] [,67] [,68] [,69] [,70] [,71] [,72] [,73] [,74]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 4 4
## [5,] 7 7 8 8 8 8 8 8 8 8 8 8
## [6,] 11 12 9 9 9 9 10 10 10 11 11 12
## [7,] 13 13 10 11 12 13 11 12 13 12 13 13
## [,75] [,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [,86]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 4 4 4 4 4 4 4 4 4 4 5 5
## [5,] 9 9 9 9 9 9 10 10 10 11 6 6
## [6,] 10 10 10 11 11 12 11 11 12 12 7 7
## [7,] 11 12 13 12 13 13 12 13 13 13 8 9
## [,87] [,88] [,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98]
## [1,] 1 1 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 6 6 6 6 6
## [6,] 7 7 7 7 8 8 8 8 8 9 9 9
## [7,] 10 11 12 13 9 10 11 12 13 10 11 12
## [,99] [,100] [,101] [,102] [,103] [,104] [,105] [,106] [,107] [,108]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 7 7 7
## [6,] 9 10 10 10 11 11 12 8 8 8
## [7,] 13 11 12 13 12 13 13 9 10 11
## [,109] [,110] [,111] [,112] [,113] [,114] [,115] [,116] [,117] [,118]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 9 9 9 9 10 10 10 11
## [7,] 12 13 10 11 12 13 11 12 13 12
## [,119] [,120] [,121] [,122] [,123] [,124] [,125] [,126] [,127] [,128]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 8 8 8 8 8 8 8 8
## [6,] 11 12 9 9 9 9 10 10 10 11
## [7,] 13 13 10 11 12 13 11 12 13 12
## [,129] [,130] [,131] [,132] [,133] [,134] [,135] [,136] [,137] [,138]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 8 8 9 9 9 9 9 9 10 10
## [6,] 11 12 10 10 10 11 11 12 11 11
## [7,] 13 13 11 12 13 12 13 13 12 13
## [,139] [,140] [,141] [,142] [,143] [,144] [,145] [,146] [,147] [,148]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 5 5 6 6 6 6 6 6 6 6
## [5,] 10 11 7 7 7 7 7 7 7 7
## [6,] 12 12 8 8 8 8 8 9 9 9
## [7,] 13 13 9 10 11 12 13 10 11 12
## [,149] [,150] [,151] [,152] [,153] [,154] [,155] [,156] [,157] [,158]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 8 8 8
## [6,] 9 10 10 10 11 11 12 9 9 9
## [7,] 13 11 12 13 12 13 13 10 11 12
## [,159] [,160] [,161] [,162] [,163] [,164] [,165] [,166] [,167] [,168]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 8 8 9 9 9
## [6,] 9 10 10 10 11 11 12 10 10 10
## [7,] 13 11 12 13 12 13 13 11 12 13
## [,169] [,170] [,171] [,172] [,173] [,174] [,175] [,176] [,177] [,178]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 6 6 6 6 6 6 6 7 7 7
## [5,] 9 9 9 10 10 10 11 8 8 8
## [6,] 11 11 12 11 11 12 12 9 9 9
## [7,] 12 13 13 12 13 13 13 10 11 12
## [,179] [,180] [,181] [,182] [,183] [,184] [,185] [,186] [,187] [,188]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 9 9 9
## [6,] 9 10 10 10 11 11 12 10 10 10
## [7,] 13 11 12 13 12 13 13 11 12 13
## [,189] [,190] [,191] [,192] [,193] [,194] [,195] [,196] [,197] [,198]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 7 7 7 7 7 7 7 8 8 8
## [5,] 9 9 9 10 10 10 11 9 9 9
## [6,] 11 11 12 11 11 12 12 10 10 10
## [7,] 12 13 13 12 13 13 13 11 12 13
## [,199] [,200] [,201] [,202] [,203] [,204] [,205] [,206] [,207] [,208]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3
## [4,] 8 8 8 8 8 8 8 9 9 9
## [5,] 9 9 9 10 10 10 11 10 10 10
## [6,] 11 11 12 11 11 12 12 11 11 12
## [7,] 12 13 13 12 13 13 13 12 13 13
## [,209] [,210] [,211] [,212] [,213] [,214] [,215] [,216] [,217] [,218]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 3 3 4 4 4 4 4 4 4 4
## [4,] 9 10 5 5 5 5 5 5 5 5
## [5,] 11 11 6 6 6 6 6 6 6 6
## [6,] 12 12 7 7 7 7 7 7 8 8
## [7,] 13 13 8 9 10 11 12 13 9 10
## [,219] [,220] [,221] [,222] [,223] [,224] [,225] [,226] [,227] [,228]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 6 6 6
## [6,] 8 8 8 9 9 9 9 10 10 10
## [7,] 11 12 13 10 11 12 13 11 12 13
## [,229] [,230] [,231] [,232] [,233] [,234] [,235] [,236] [,237] [,238]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 7 7 7 7 7 7 7
## [6,] 11 11 12 8 8 8 8 8 9 9
## [7,] 12 13 13 9 10 11 12 13 10 11
## [,239] [,240] [,241] [,242] [,243] [,244] [,245] [,246] [,247] [,248]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 7 7 7 7 7 7 8 8
## [6,] 9 9 10 10 10 11 11 12 9 9
## [7,] 12 13 11 12 13 12 13 13 10 11
## [,249] [,250] [,251] [,252] [,253] [,254] [,255] [,256] [,257] [,258]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 8 8 8 8 8 8 8 8 9 9
## [6,] 9 9 10 10 10 11 11 12 10 10
## [7,] 12 13 11 12 13 12 13 13 11 12
## [,259] [,260] [,261] [,262] [,263] [,264] [,265] [,266] [,267] [,268]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 6 6
## [5,] 9 9 9 9 10 10 10 11 7 7
## [6,] 10 11 11 12 11 11 12 12 8 8
## [7,] 13 12 13 13 12 13 13 13 9 10
## [,269] [,270] [,271] [,272] [,273] [,274] [,275] [,276] [,277] [,278]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 9 9 9 9 10 10 10
## [7,] 11 12 13 10 11 12 13 11 12 13
## [,279] [,280] [,281] [,282] [,283] [,284] [,285] [,286] [,287] [,288]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 8 8 8 8 8 8 8
## [6,] 11 11 12 9 9 9 9 10 10 10
## [7,] 12 13 13 10 11 12 13 11 12 13
## [,289] [,290] [,291] [,292] [,293] [,294] [,295] [,296] [,297] [,298]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 9 9 9 9 9 9 10
## [6,] 11 11 12 10 10 10 11 11 12 11
## [7,] 12 13 13 11 12 13 12 13 13 12
## [,299] [,300] [,301] [,302] [,303] [,304] [,305] [,306] [,307] [,308]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 7 7 7 7 7 7 7
## [5,] 10 10 11 8 8 8 8 8 8 8
## [6,] 11 12 12 9 9 9 9 10 10 10
## [7,] 13 13 13 10 11 12 13 11 12 13
## [,309] [,310] [,311] [,312] [,313] [,314] [,315] [,316] [,317] [,318]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 9 9 9 9 9 9 10
## [6,] 11 11 12 10 10 10 11 11 12 11
## [7,] 12 13 13 11 12 13 12 13 13 12
## [,319] [,320] [,321] [,322] [,323] [,324] [,325] [,326] [,327] [,328]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 8 8 8 8 8 8 8
## [5,] 10 10 11 9 9 9 9 9 9 10
## [6,] 11 12 12 10 10 10 11 11 12 11
## [7,] 13 13 13 11 12 13 12 13 13 12
## [,329] [,330] [,331] [,332] [,333] [,334] [,335] [,336] [,337] [,338]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 4 4 4 4 4 4 4 4 5 5
## [4,] 8 8 8 9 9 9 9 10 6 6
## [5,] 10 10 11 10 10 10 11 11 7 7
## [6,] 11 12 12 11 11 12 12 12 8 8
## [7,] 13 13 13 12 13 13 13 13 9 10
## [,339] [,340] [,341] [,342] [,343] [,344] [,345] [,346] [,347] [,348]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 9 9 9 9 10 10 10
## [7,] 11 12 13 10 11 12 13 11 12 13
## [,349] [,350] [,351] [,352] [,353] [,354] [,355] [,356] [,357] [,358]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 8 8 8 8 8 8 8
## [6,] 11 11 12 9 9 9 9 10 10 10
## [7,] 12 13 13 10 11 12 13 11 12 13
## [,359] [,360] [,361] [,362] [,363] [,364] [,365] [,366] [,367] [,368]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 9 9 9 9 9 9 10
## [6,] 11 11 12 10 10 10 11 11 12 11
## [7,] 12 13 13 11 12 13 12 13 13 12
## [,369] [,370] [,371] [,372] [,373] [,374] [,375] [,376] [,377] [,378]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 7 7 7 7 7 7 7
## [5,] 10 10 11 8 8 8 8 8 8 8
## [6,] 11 12 12 9 9 9 9 10 10 10
## [7,] 13 13 13 10 11 12 13 11 12 13
## [,379] [,380] [,381] [,382] [,383] [,384] [,385] [,386] [,387] [,388]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 9 9 9 9 9 9 10
## [6,] 11 11 12 10 10 10 11 11 12 11
## [7,] 12 13 13 11 12 13 12 13 13 12
## [,389] [,390] [,391] [,392] [,393] [,394] [,395] [,396] [,397] [,398]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 8 8 8 8 8 8 8
## [5,] 10 10 11 9 9 9 9 9 9 10
## [6,] 11 12 12 10 10 10 11 11 12 11
## [7,] 13 13 13 11 12 13 12 13 13 12
## [,399] [,400] [,401] [,402] [,403] [,404] [,405] [,406] [,407] [,408]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 5 5 5 5 5 5 5 5 6 6
## [4,] 8 8 8 9 9 9 9 10 7 7
## [5,] 10 10 11 10 10 10 11 11 8 8
## [6,] 11 12 12 11 11 12 12 12 9 9
## [7,] 13 13 13 12 13 13 13 13 10 11
## [,409] [,410] [,411] [,412] [,413] [,414] [,415] [,416] [,417] [,418]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 9 9
## [6,] 9 9 10 10 10 11 11 12 10 10
## [7,] 12 13 11 12 13 12 13 13 11 12
## [,419] [,420] [,421] [,422] [,423] [,424] [,425] [,426] [,427] [,428]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 8 8
## [5,] 9 9 9 9 10 10 10 11 9 9
## [6,] 10 11 11 12 11 11 12 12 10 10
## [7,] 13 12 13 13 12 13 13 13 11 12
## [,429] [,430] [,431] [,432] [,433] [,434] [,435] [,436] [,437] [,438]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 8 9 9
## [5,] 9 9 9 9 10 10 10 11 10 10
## [6,] 10 11 11 12 11 11 12 12 11 11
## [7,] 13 12 13 13 12 13 13 13 12 13
## [,439] [,440] [,441] [,442] [,443] [,444] [,445] [,446] [,447] [,448]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 6 6 6 7 7 7 7 7 7 7
## [4,] 9 9 10 8 8 8 8 8 8 8
## [5,] 10 11 11 9 9 9 9 9 9 10
## [6,] 12 12 12 10 10 10 11 11 12 11
## [7,] 13 13 13 11 12 13 12 13 13 12
## [,449] [,450] [,451] [,452] [,453] [,454] [,455] [,456] [,457] [,458]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 2 2 2 2 2 2
## [3,] 7 7 7 7 7 7 7 7 8 8
## [4,] 8 8 8 9 9 9 9 10 9 9
## [5,] 10 10 11 10 10 10 11 11 10 10
## [6,] 11 12 12 11 11 12 12 12 11 11
## [7,] 13 13 13 12 13 13 13 13 12 13
## [,459] [,460] [,461] [,462] [,463] [,464] [,465] [,466] [,467] [,468]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 2 2 2 2 3 3 3 3 3 3
## [3,] 8 8 8 9 4 4 4 4 4 4
## [4,] 9 9 10 10 5 5 5 5 5 5
## [5,] 10 11 11 11 6 6 6 6 6 6
## [6,] 12 12 12 12 7 7 7 7 7 7
## [7,] 13 13 13 13 8 9 10 11 12 13
## [,469] [,470] [,471] [,472] [,473] [,474] [,475] [,476] [,477] [,478]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 6 6 6
## [6,] 8 8 8 8 8 9 9 9 9 10
## [7,] 9 10 11 12 13 10 11 12 13 11
## [,479] [,480] [,481] [,482] [,483] [,484] [,485] [,486] [,487] [,488]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 7 7 7 7 7
## [6,] 10 10 11 11 12 8 8 8 8 8
## [7,] 12 13 12 13 13 9 10 11 12 13
## [,489] [,490] [,491] [,492] [,493] [,494] [,495] [,496] [,497] [,498]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 9 9 9 9 10 10 10 11 11 12
## [7,] 10 11 12 13 11 12 13 12 13 13
## [,499] [,500] [,501] [,502] [,503] [,504] [,505] [,506] [,507] [,508]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 8 8 8 8 8 8 8 8 8 8
## [6,] 9 9 9 9 10 10 10 11 11 12
## [7,] 10 11 12 13 11 12 13 12 13 13
## [,509] [,510] [,511] [,512] [,513] [,514] [,515] [,516] [,517] [,518]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 9 9 9 9 9 9 10 10 10 11
## [6,] 10 10 10 11 11 12 11 11 12 12
## [7,] 11 12 13 12 13 13 12 13 13 13
## [,519] [,520] [,521] [,522] [,523] [,524] [,525] [,526] [,527] [,528]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 8 8 9 9 9 9 10
## [7,] 9 10 11 12 13 10 11 12 13 11
## [,529] [,530] [,531] [,532] [,533] [,534] [,535] [,536] [,537] [,538]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 8 8 8 8 8
## [6,] 10 10 11 11 12 9 9 9 9 10
## [7,] 12 13 12 13 13 10 11 12 13 11
## [,539] [,540] [,541] [,542] [,543] [,544] [,545] [,546] [,547] [,548]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 9 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11 11
## [7,] 12 13 12 13 13 11 12 13 12 13
## [,549] [,550] [,551] [,552] [,553] [,554] [,555] [,556] [,557] [,558]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 7 7 7 7 7
## [5,] 9 10 10 10 11 8 8 8 8 8
## [6,] 12 11 11 12 12 9 9 9 9 10
## [7,] 13 12 13 13 13 10 11 12 13 11
## [,559] [,560] [,561] [,562] [,563] [,564] [,565] [,566] [,567] [,568]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 9 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11 11
## [7,] 12 13 12 13 13 11 12 13 12 13
## [,569] [,570] [,571] [,572] [,573] [,574] [,575] [,576] [,577] [,578]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 7 7 8 8 8 8 8
## [5,] 9 10 10 10 11 9 9 9 9 9
## [6,] 12 11 11 12 12 10 10 10 11 11
## [7,] 13 12 13 13 13 11 12 13 12 13
## [,579] [,580] [,581] [,582] [,583] [,584] [,585] [,586] [,587] [,588]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 8 8 8 8 8 9 9 9 9 10
## [5,] 9 10 10 10 11 10 10 10 11 11
## [6,] 12 11 11 12 12 11 11 12 12 12
## [7,] 13 12 13 13 13 12 13 13 13 13
## [,589] [,590] [,591] [,592] [,593] [,594] [,595] [,596] [,597] [,598]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 8 8 9 9 9 9 10
## [7,] 9 10 11 12 13 10 11 12 13 11
## [,599] [,600] [,601] [,602] [,603] [,604] [,605] [,606] [,607] [,608]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 8 8 8 8 8
## [6,] 10 10 11 11 12 9 9 9 9 10
## [7,] 12 13 12 13 13 10 11 12 13 11
## [,609] [,610] [,611] [,612] [,613] [,614] [,615] [,616] [,617] [,618]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 9 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11 11
## [7,] 12 13 12 13 13 11 12 13 12 13
## [,619] [,620] [,621] [,622] [,623] [,624] [,625] [,626] [,627] [,628]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 7 7 7 7 7
## [5,] 9 10 10 10 11 8 8 8 8 8
## [6,] 12 11 11 12 12 9 9 9 9 10
## [7,] 13 12 13 13 13 10 11 12 13 11
## [,629] [,630] [,631] [,632] [,633] [,634] [,635] [,636] [,637] [,638]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 9 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11 11
## [7,] 12 13 12 13 13 11 12 13 12 13
## [,639] [,640] [,641] [,642] [,643] [,644] [,645] [,646] [,647] [,648]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 8 8 8 8 8
## [5,] 9 10 10 10 11 9 9 9 9 9
## [6,] 12 11 11 12 12 10 10 10 11 11
## [7,] 13 12 13 13 13 11 12 13 12 13
## [,649] [,650] [,651] [,652] [,653] [,654] [,655] [,656] [,657] [,658]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 8 8 8 8 8 9 9 9 9 10
## [5,] 9 10 10 10 11 10 10 10 11 11
## [6,] 12 11 11 12 12 11 11 12 12 12
## [7,] 13 12 13 13 13 12 13 13 13 13
## [,659] [,660] [,661] [,662] [,663] [,664] [,665] [,666] [,667] [,668]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 8 8
## [6,] 9 9 9 9 10 10 10 11 11 12
## [7,] 10 11 12 13 11 12 13 12 13 13
## [,669] [,670] [,671] [,672] [,673] [,674] [,675] [,676] [,677] [,678]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 9 9 9 9 9 9 10 10 10 11
## [6,] 10 10 10 11 11 12 11 11 12 12
## [7,] 11 12 13 12 13 13 12 13 13 13
## [,679] [,680] [,681] [,682] [,683] [,684] [,685] [,686] [,687] [,688]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 9 10 10 10 11
## [6,] 10 10 10 11 11 12 11 11 12 12
## [7,] 11 12 13 12 13 13 12 13 13 13
## [,689] [,690] [,691] [,692] [,693] [,694] [,695] [,696] [,697] [,698]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 7 7 7 7 7
## [4,] 9 9 9 9 10 8 8 8 8 8
## [5,] 10 10 10 11 11 9 9 9 9 9
## [6,] 11 11 12 12 12 10 10 10 11 11
## [7,] 12 13 13 13 13 11 12 13 12 13
## [,699] [,700] [,701] [,702] [,703] [,704] [,705] [,706] [,707] [,708]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 7 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 9 9 9 9 10
## [5,] 9 10 10 10 11 10 10 10 11 11
## [6,] 12 11 11 12 12 11 11 12 12 12
## [7,] 13 12 13 13 13 12 13 13 13 13
## [,709] [,710] [,711] [,712] [,713] [,714] [,715] [,716] [,717] [,718]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 3 3 3 3 3 3 4 4 4 4
## [3,] 8 8 8 8 8 9 5 5 5 5
## [4,] 9 9 9 9 10 10 6 6 6 6
## [5,] 10 10 10 11 11 11 7 7 7 7
## [6,] 11 11 12 12 12 12 8 8 8 8
## [7,] 12 13 13 13 13 13 9 10 11 12
## [,719] [,720] [,721] [,722] [,723] [,724] [,725] [,726] [,727] [,728]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 9 9 9 9 10 10 10 11 11
## [7,] 13 10 11 12 13 11 12 13 12 13
## [,729] [,730] [,731] [,732] [,733] [,734] [,735] [,736] [,737] [,738]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 8 8 8 8 8 8 8 8 8
## [6,] 12 9 9 9 9 10 10 10 11 11
## [7,] 13 10 11 12 13 11 12 13 12 13
## [,739] [,740] [,741] [,742] [,743] [,744] [,745] [,746] [,747] [,748]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 8 9 9 9 9 9 9 10 10 10
## [6,] 12 10 10 10 11 11 12 11 11 12
## [7,] 13 11 12 13 12 13 13 12 13 13
## [,749] [,750] [,751] [,752] [,753] [,754] [,755] [,756] [,757] [,758]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 6 7 7 7 7 7 7 7 7 7
## [5,] 11 8 8 8 8 8 8 8 8 8
## [6,] 12 9 9 9 9 10 10 10 11 11
## [7,] 13 10 11 12 13 11 12 13 12 13
## [,759] [,760] [,761] [,762] [,763] [,764] [,765] [,766] [,767] [,768]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 9 9 9 9 9 9 10 10 10
## [6,] 12 10 10 10 11 11 12 11 11 12
## [7,] 13 11 12 13 12 13 13 12 13 13
## [,769] [,770] [,771] [,772] [,773] [,774] [,775] [,776] [,777] [,778]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5 5
## [4,] 7 8 8 8 8 8 8 8 8 8
## [5,] 11 9 9 9 9 9 9 10 10 10
## [6,] 12 10 10 10 11 11 12 11 11 12
## [7,] 13 11 12 13 12 13 13 12 13 13
## [,779] [,780] [,781] [,782] [,783] [,784] [,785] [,786] [,787] [,788]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 6 6 6 6
## [4,] 8 9 9 9 9 10 7 7 7 7
## [5,] 11 10 10 10 11 11 8 8 8 8
## [6,] 12 11 11 12 12 12 9 9 9 9
## [7,] 13 12 13 13 13 13 10 11 12 13
## [,789] [,790] [,791] [,792] [,793] [,794] [,795] [,796] [,797] [,798]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 9 9 9 9
## [6,] 10 10 10 11 11 12 10 10 10 11
## [7,] 11 12 13 12 13 13 11 12 13 12
## [,799] [,800] [,801] [,802] [,803] [,804] [,805] [,806] [,807] [,808]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 8 8 8 8
## [5,] 9 9 10 10 10 11 9 9 9 9
## [6,] 11 12 11 11 12 12 10 10 10 11
## [7,] 13 13 12 13 13 13 11 12 13 12
## [,809] [,810] [,811] [,812] [,813] [,814] [,815] [,816] [,817] [,818]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 9 9 9 9
## [5,] 9 9 10 10 10 11 10 10 10 11
## [6,] 11 12 11 11 12 12 11 11 12 12
## [7,] 13 13 12 13 13 13 12 13 13 13
## [,819] [,820] [,821] [,822] [,823] [,824] [,825] [,826] [,827] [,828]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 6 7 7 7 7 7 7 7 7 7
## [4,] 10 8 8 8 8 8 8 8 8 8
## [5,] 11 9 9 9 9 9 9 10 10 10
## [6,] 12 10 10 10 11 11 12 11 11 12
## [7,] 13 11 12 13 12 13 13 12 13 13
## [,829] [,830] [,831] [,832] [,833] [,834] [,835] [,836] [,837] [,838]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 4 4 4 4 4 4 4 4
## [3,] 7 7 7 7 7 7 8 8 8 8
## [4,] 8 9 9 9 9 10 9 9 9 9
## [5,] 11 10 10 10 11 11 10 10 10 11
## [6,] 12 11 11 12 12 12 11 11 12 12
## [7,] 13 12 13 13 13 13 12 13 13 13
## [,839] [,840] [,841] [,842] [,843] [,844] [,845] [,846] [,847] [,848]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 4 4 5 5 5 5 5 5 5 5
## [3,] 8 9 6 6 6 6 6 6 6 6
## [4,] 10 10 7 7 7 7 7 7 7 7
## [5,] 11 11 8 8 8 8 8 8 8 8
## [6,] 12 12 9 9 9 9 10 10 10 11
## [7,] 13 13 10 11 12 13 11 12 13 12
## [,849] [,850] [,851] [,852] [,853] [,854] [,855] [,856] [,857] [,858]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 5 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7 7
## [5,] 8 8 9 9 9 9 9 9 10 10
## [6,] 11 12 10 10 10 11 11 12 11 11
## [7,] 13 13 11 12 13 12 13 13 12 13
## [,859] [,860] [,861] [,862] [,863] [,864] [,865] [,866] [,867] [,868]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 5 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6 6
## [4,] 7 7 8 8 8 8 8 8 8 8
## [5,] 10 11 9 9 9 9 9 9 10 10
## [6,] 12 12 10 10 10 11 11 12 11 11
## [7,] 13 13 11 12 13 12 13 13 12 13
## [,869] [,870] [,871] [,872] [,873] [,874] [,875] [,876] [,877] [,878]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 5 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 7 7 7
## [4,] 8 8 9 9 9 9 10 8 8 8
## [5,] 10 11 10 10 10 11 11 9 9 9
## [6,] 12 12 11 11 12 12 12 10 10 10
## [7,] 13 13 12 13 13 13 13 11 12 13
## [,879] [,880] [,881] [,882] [,883] [,884] [,885] [,886] [,887] [,888]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 5 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 9 9 9
## [5,] 9 9 9 10 10 10 11 10 10 10
## [6,] 11 11 12 11 11 12 12 11 11 12
## [7,] 12 13 13 12 13 13 13 12 13 13
## [,889] [,890] [,891] [,892] [,893] [,894] [,895] [,896] [,897] [,898]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 5 5 5 5 5 5 5 5 6 6
## [3,] 7 7 8 8 8 8 8 9 7 7
## [4,] 9 10 9 9 9 9 10 10 8 8
## [5,] 11 11 10 10 10 11 11 11 9 9
## [6,] 12 12 11 11 12 12 12 12 10 10
## [7,] 13 13 12 13 13 13 13 13 11 12
## [,899] [,900] [,901] [,902] [,903] [,904] [,905] [,906] [,907] [,908]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 6 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 8 9 9
## [5,] 9 9 9 9 10 10 10 11 10 10
## [6,] 10 11 11 12 11 11 12 12 11 11
## [7,] 13 12 13 13 12 13 13 13 12 13
## [,909] [,910] [,911] [,912] [,913] [,914] [,915] [,916] [,917] [,918]
## [1,] 1 1 1 1 1 1 1 1 1 1
## [2,] 6 6 6 6 6 6 6 6 6 7
## [3,] 7 7 7 8 8 8 8 8 9 8
## [4,] 9 9 10 9 9 9 9 10 10 9
## [5,] 10 11 11 10 10 10 11 11 11 10
## [6,] 12 12 12 11 11 12 12 12 12 11
## [7,] 13 13 13 12 13 13 13 13 13 12
## [,919] [,920] [,921] [,922] [,923] [,924] [,925] [,926] [,927] [,928]
## [1,] 1 1 1 1 1 1 2 2 2 2
## [2,] 7 7 7 7 7 8 3 3 3 3
## [3,] 8 8 8 8 9 9 4 4 4 4
## [4,] 9 9 9 10 10 10 5 5 5 5
## [5,] 10 10 11 11 11 11 6 6 6 6
## [6,] 11 12 12 12 12 12 7 7 7 7
## [7,] 13 13 13 13 13 13 8 9 10 11
## [,929] [,930] [,931] [,932] [,933] [,934] [,935] [,936] [,937] [,938]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 6 6 6
## [6,] 7 7 8 8 8 8 8 9 9 9
## [7,] 12 13 9 10 11 12 13 10 11 12
## [,939] [,940] [,941] [,942] [,943] [,944] [,945] [,946] [,947] [,948]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 6 6 6 6 6 6 6 7 7 7
## [6,] 9 10 10 10 11 11 12 8 8 8
## [7,] 13 11 12 13 12 13 13 9 10 11
## [,949] [,950] [,951] [,952] [,953] [,954] [,955] [,956] [,957] [,958]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 7 7 7 7 7 7 7 7
## [6,] 8 8 9 9 9 9 10 10 10 11
## [7,] 12 13 10 11 12 13 11 12 13 12
## [,959] [,960] [,961] [,962] [,963] [,964] [,965] [,966] [,967] [,968]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 7 7 8 8 8 8 8 8 8 8
## [6,] 11 12 9 9 9 9 10 10 10 11
## [7,] 13 13 10 11 12 13 11 12 13 12
## [,969] [,970] [,971] [,972] [,973] [,974] [,975] [,976] [,977] [,978]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 5 5 5 5 5 5 5 5
## [5,] 8 8 9 9 9 9 9 9 10 10
## [6,] 11 12 10 10 10 11 11 12 11 11
## [7,] 13 13 11 12 13 12 13 13 12 13
## [,979] [,980] [,981] [,982] [,983] [,984] [,985] [,986] [,987] [,988]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 5 5 6 6 6 6 6 6 6 6
## [5,] 10 11 7 7 7 7 7 7 7 7
## [6,] 12 12 8 8 8 8 8 9 9 9
## [7,] 13 13 9 10 11 12 13 10 11 12
## [,989] [,990] [,991] [,992] [,993] [,994] [,995] [,996] [,997] [,998]
## [1,] 2 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 8 8 8
## [6,] 9 10 10 10 11 11 12 9 9 9
## [7,] 13 11 12 13 12 13 13 10 11 12
## [,999] [,1000] [,1001] [,1002] [,1003] [,1004] [,1005] [,1006] [,1007]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 8 8 9 9
## [6,] 9 10 10 10 11 11 12 10 10
## [7,] 13 11 12 13 12 13 13 11 12
## [,1008] [,1009] [,1010] [,1011] [,1012] [,1013] [,1014] [,1015] [,1016]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4
## [4,] 6 6 6 6 6 6 6 6 7
## [5,] 9 9 9 9 10 10 10 11 8
## [6,] 10 11 11 12 11 11 12 12 9
## [7,] 13 12 13 13 12 13 13 13 10
## [,1017] [,1018] [,1019] [,1020] [,1021] [,1022] [,1023] [,1024] [,1025]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 8
## [6,] 9 9 9 10 10 10 11 11 12
## [7,] 11 12 13 11 12 13 12 13 13
## [,1026] [,1027] [,1028] [,1029] [,1030] [,1031] [,1032] [,1033] [,1034]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 9 9 9 9 9 9 10 10 10
## [6,] 10 10 10 11 11 12 11 11 12
## [7,] 11 12 13 12 13 13 12 13 13
## [,1035] [,1036] [,1037] [,1038] [,1039] [,1040] [,1041] [,1042] [,1043]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 4 4
## [4,] 7 8 8 8 8 8 8 8 8
## [5,] 11 9 9 9 9 9 9 10 10
## [6,] 12 10 10 10 11 11 12 11 11
## [7,] 13 11 12 13 12 13 13 12 13
## [,1044] [,1045] [,1046] [,1047] [,1048] [,1049] [,1050] [,1051] [,1052]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 4 4 4 4 4 4 4 5 5
## [4,] 8 8 9 9 9 9 10 6 6
## [5,] 10 11 10 10 10 11 11 7 7
## [6,] 12 12 11 11 12 12 12 8 8
## [7,] 13 13 12 13 13 13 13 9 10
## [,1053] [,1054] [,1055] [,1056] [,1057] [,1058] [,1059] [,1060] [,1061]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 9 9 9 9 10 10
## [7,] 11 12 13 10 11 12 13 11 12
## [,1062] [,1063] [,1064] [,1065] [,1066] [,1067] [,1068] [,1069] [,1070]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 8 8 8 8 8
## [6,] 10 11 11 12 9 9 9 9 10
## [7,] 13 12 13 13 10 11 12 13 11
## [,1071] [,1072] [,1073] [,1074] [,1075] [,1076] [,1077] [,1078] [,1079]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11
## [7,] 12 13 12 13 13 11 12 13 12
## [,1080] [,1081] [,1082] [,1083] [,1084] [,1085] [,1086] [,1087] [,1088]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 7 7 7
## [5,] 9 9 10 10 10 11 8 8 8
## [6,] 11 12 11 11 12 12 9 9 9
## [7,] 13 13 12 13 13 13 10 11 12
## [,1089] [,1090] [,1091] [,1092] [,1093] [,1094] [,1095] [,1096] [,1097]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 9 9
## [6,] 9 10 10 10 11 11 12 10 10
## [7,] 13 11 12 13 12 13 13 11 12
## [,1098] [,1099] [,1100] [,1101] [,1102] [,1103] [,1104] [,1105] [,1106]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 8
## [5,] 9 9 9 9 10 10 10 11 9
## [6,] 10 11 11 12 11 11 12 12 10
## [7,] 13 12 13 13 12 13 13 13 11
## [,1107] [,1108] [,1109] [,1110] [,1111] [,1112] [,1113] [,1114] [,1115]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 10 10 10 11
## [6,] 10 10 11 11 12 11 11 12 12
## [7,] 12 13 12 13 13 12 13 13 13
## [,1116] [,1117] [,1118] [,1119] [,1120] [,1121] [,1122] [,1123] [,1124]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 5 5 5 5 5 6 6 6 6
## [4,] 9 9 9 9 10 7 7 7 7
## [5,] 10 10 10 11 11 8 8 8 8
## [6,] 11 11 12 12 12 9 9 9 9
## [7,] 12 13 13 13 13 10 11 12 13
## [,1125] [,1126] [,1127] [,1128] [,1129] [,1130] [,1131] [,1132] [,1133]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 9 9 9
## [6,] 10 10 10 11 11 12 10 10 10
## [7,] 11 12 13 12 13 13 11 12 13
## [,1134] [,1135] [,1136] [,1137] [,1138] [,1139] [,1140] [,1141] [,1142]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 8 8
## [5,] 9 9 9 10 10 10 11 9 9
## [6,] 11 11 12 11 11 12 12 10 10
## [7,] 12 13 13 12 13 13 13 11 12
## [,1143] [,1144] [,1145] [,1146] [,1147] [,1148] [,1149] [,1150] [,1151]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 8 9
## [5,] 9 9 9 9 10 10 10 11 10
## [6,] 10 11 11 12 11 11 12 12 11
## [7,] 13 12 13 13 12 13 13 13 12
## [,1152] [,1153] [,1154] [,1155] [,1156] [,1157] [,1158] [,1159] [,1160]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 6 6 6 6 7 7 7 7 7
## [4,] 9 9 9 10 8 8 8 8 8
## [5,] 10 10 11 11 9 9 9 9 9
## [6,] 11 12 12 12 10 10 10 11 11
## [7,] 13 13 13 13 11 12 13 12 13
## [,1161] [,1162] [,1163] [,1164] [,1165] [,1166] [,1167] [,1168] [,1169]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 9 9 9 9
## [5,] 9 10 10 10 11 10 10 10 11
## [6,] 12 11 11 12 12 11 11 12 12
## [7,] 13 12 13 13 13 12 13 13 13
## [,1170] [,1171] [,1172] [,1173] [,1174] [,1175] [,1176] [,1177] [,1178]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 3 3 3 3 3 3 3 4 4
## [3,] 7 8 8 8 8 8 9 5 5
## [4,] 10 9 9 9 9 10 10 6 6
## [5,] 11 10 10 10 11 11 11 7 7
## [6,] 12 11 11 12 12 12 12 8 8
## [7,] 13 12 13 13 13 13 13 9 10
## [,1179] [,1180] [,1181] [,1182] [,1183] [,1184] [,1185] [,1186] [,1187]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 7 7
## [6,] 8 8 8 9 9 9 9 10 10
## [7,] 11 12 13 10 11 12 13 11 12
## [,1188] [,1189] [,1190] [,1191] [,1192] [,1193] [,1194] [,1195] [,1196]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 8 8 8 8 8
## [6,] 10 11 11 12 9 9 9 9 10
## [7,] 13 12 13 13 10 11 12 13 11
## [,1197] [,1198] [,1199] [,1200] [,1201] [,1202] [,1203] [,1204] [,1205]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11
## [7,] 12 13 12 13 13 11 12 13 12
## [,1206] [,1207] [,1208] [,1209] [,1210] [,1211] [,1212] [,1213] [,1214]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 7 7 7
## [5,] 9 9 10 10 10 11 8 8 8
## [6,] 11 12 11 11 12 12 9 9 9
## [7,] 13 13 12 13 13 13 10 11 12
## [,1215] [,1216] [,1217] [,1218] [,1219] [,1220] [,1221] [,1222] [,1223]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 9 9
## [6,] 9 10 10 10 11 11 12 10 10
## [7,] 13 11 12 13 12 13 13 11 12
## [,1224] [,1225] [,1226] [,1227] [,1228] [,1229] [,1230] [,1231] [,1232]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 8
## [5,] 9 9 9 9 10 10 10 11 9
## [6,] 10 11 11 12 11 11 12 12 10
## [7,] 13 12 13 13 12 13 13 13 11
## [,1233] [,1234] [,1235] [,1236] [,1237] [,1238] [,1239] [,1240] [,1241]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 10 10 10 11
## [6,] 10 10 11 11 12 11 11 12 12
## [7,] 12 13 12 13 13 12 13 13 13
## [,1242] [,1243] [,1244] [,1245] [,1246] [,1247] [,1248] [,1249] [,1250]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 6 6 6 6
## [4,] 9 9 9 9 10 7 7 7 7
## [5,] 10 10 10 11 11 8 8 8 8
## [6,] 11 11 12 12 12 9 9 9 9
## [7,] 12 13 13 13 13 10 11 12 13
## [,1251] [,1252] [,1253] [,1254] [,1255] [,1256] [,1257] [,1258] [,1259]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 9 9 9
## [6,] 10 10 10 11 11 12 10 10 10
## [7,] 11 12 13 12 13 13 11 12 13
## [,1260] [,1261] [,1262] [,1263] [,1264] [,1265] [,1266] [,1267] [,1268]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 8 8
## [5,] 9 9 9 10 10 10 11 9 9
## [6,] 11 11 12 11 11 12 12 10 10
## [7,] 12 13 13 12 13 13 13 11 12
## [,1269] [,1270] [,1271] [,1272] [,1273] [,1274] [,1275] [,1276] [,1277]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 8 9
## [5,] 9 9 9 9 10 10 10 11 10
## [6,] 10 11 11 12 11 11 12 12 11
## [7,] 13 12 13 13 12 13 13 13 12
## [,1278] [,1279] [,1280] [,1281] [,1282] [,1283] [,1284] [,1285] [,1286]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 7 7 7 7 7
## [4,] 9 9 9 10 8 8 8 8 8
## [5,] 10 10 11 11 9 9 9 9 9
## [6,] 11 12 12 12 10 10 10 11 11
## [7,] 13 13 13 13 11 12 13 12 13
## [,1287] [,1288] [,1289] [,1290] [,1291] [,1292] [,1293] [,1294] [,1295]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 9 9 9 9
## [5,] 9 10 10 10 11 10 10 10 11
## [6,] 12 11 11 12 12 11 11 12 12
## [7,] 13 12 13 13 13 12 13 13 13
## [,1296] [,1297] [,1298] [,1299] [,1300] [,1301] [,1302] [,1303] [,1304]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 4 4 4 4 4 4 4 5 5
## [3,] 7 8 8 8 8 8 9 6 6
## [4,] 10 9 9 9 9 10 10 7 7
## [5,] 11 10 10 10 11 11 11 8 8
## [6,] 12 11 11 12 12 12 12 9 9
## [7,] 13 12 13 13 13 13 13 10 11
## [,1305] [,1306] [,1307] [,1308] [,1309] [,1310] [,1311] [,1312] [,1313]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 9
## [6,] 9 9 10 10 10 11 11 12 10
## [7,] 12 13 11 12 13 12 13 13 11
## [,1314] [,1315] [,1316] [,1317] [,1318] [,1319] [,1320] [,1321] [,1322]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 9 9 9 9 9 10 10 10 11
## [6,] 10 10 11 11 12 11 11 12 12
## [7,] 12 13 12 13 13 12 13 13 13
## [,1323] [,1324] [,1325] [,1326] [,1327] [,1328] [,1329] [,1330] [,1331]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 9 10 10 10
## [6,] 10 10 10 11 11 12 11 11 12
## [7,] 11 12 13 12 13 13 12 13 13
## [,1332] [,1333] [,1334] [,1335] [,1336] [,1337] [,1338] [,1339] [,1340]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 7 7 7
## [4,] 8 9 9 9 9 10 8 8 8
## [5,] 11 10 10 10 11 11 9 9 9
## [6,] 12 11 11 12 12 12 10 10 10
## [7,] 13 12 13 13 13 13 11 12 13
## [,1341] [,1342] [,1343] [,1344] [,1345] [,1346] [,1347] [,1348] [,1349]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 9 9
## [5,] 9 9 9 10 10 10 11 10 10
## [6,] 11 11 12 11 11 12 12 11 11
## [7,] 12 13 13 12 13 13 13 12 13
## [,1350] [,1351] [,1352] [,1353] [,1354] [,1355] [,1356] [,1357] [,1358]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 8 8 8 8 8 9
## [4,] 9 9 10 9 9 9 9 10 10
## [5,] 10 11 11 10 10 10 11 11 11
## [6,] 12 12 12 11 11 12 12 12 12
## [7,] 13 13 13 12 13 13 13 13 13
## [,1359] [,1360] [,1361] [,1362] [,1363] [,1364] [,1365] [,1366] [,1367]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 9 10 10 10
## [6,] 10 10 10 11 11 12 11 11 12
## [7,] 11 12 13 12 13 13 12 13 13
## [,1368] [,1369] [,1370] [,1371] [,1372] [,1373] [,1374] [,1375] [,1376]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 8 8 8
## [4,] 8 9 9 9 9 10 9 9 9
## [5,] 11 10 10 10 11 11 10 10 10
## [6,] 12 11 11 12 12 12 11 11 12
## [7,] 13 12 13 13 13 13 12 13 13
## [,1377] [,1378] [,1379] [,1380] [,1381] [,1382] [,1383] [,1384] [,1385]
## [1,] 2 2 2 2 2 2 2 2 2
## [2,] 6 6 6 7 7 7 7 7 7
## [3,] 8 8 9 8 8 8 8 8 9
## [4,] 9 10 10 9 9 9 9 10 10
## [5,] 11 11 11 10 10 10 11 11 11
## [6,] 12 12 12 11 11 12 12 12 12
## [7,] 13 13 13 12 13 13 13 13 13
## [,1386] [,1387] [,1388] [,1389] [,1390] [,1391] [,1392] [,1393] [,1394]
## [1,] 2 3 3 3 3 3 3 3 3
## [2,] 8 4 4 4 4 4 4 4 4
## [3,] 9 5 5 5 5 5 5 5 5
## [4,] 10 6 6 6 6 6 6 6 6
## [5,] 11 7 7 7 7 7 7 7 7
## [6,] 12 8 8 8 8 8 9 9 9
## [7,] 13 9 10 11 12 13 10 11 12
## [,1395] [,1396] [,1397] [,1398] [,1399] [,1400] [,1401] [,1402] [,1403]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 7 7 7 7 7 7 7 8 8
## [6,] 9 10 10 10 11 11 12 9 9
## [7,] 13 11 12 13 12 13 13 10 11
## [,1404] [,1405] [,1406] [,1407] [,1408] [,1409] [,1410] [,1411] [,1412]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 8 8 8 8 8 8 8 8 9
## [6,] 9 9 10 10 10 11 11 12 10
## [7,] 12 13 11 12 13 12 13 13 11
## [,1413] [,1414] [,1415] [,1416] [,1417] [,1418] [,1419] [,1420] [,1421]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 6 6 6 6 6 6 6 6 6
## [5,] 9 9 9 9 9 10 10 10 11
## [6,] 10 10 11 11 12 11 11 12 12
## [7,] 12 13 12 13 13 12 13 13 13
## [,1422] [,1423] [,1424] [,1425] [,1426] [,1427] [,1428] [,1429] [,1430]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 8
## [6,] 9 9 9 9 10 10 10 11 11
## [7,] 10 11 12 13 11 12 13 12 13
## [,1431] [,1432] [,1433] [,1434] [,1435] [,1436] [,1437] [,1438] [,1439]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 9 9 9 9 9 9 10 10
## [6,] 12 10 10 10 11 11 12 11 11
## [7,] 13 11 12 13 12 13 13 12 13
## [,1440] [,1441] [,1442] [,1443] [,1444] [,1445] [,1446] [,1447] [,1448]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 5
## [4,] 7 7 8 8 8 8 8 8 8
## [5,] 10 11 9 9 9 9 9 9 10
## [6,] 12 12 10 10 10 11 11 12 11
## [7,] 13 13 11 12 13 12 13 13 12
## [,1449] [,1450] [,1451] [,1452] [,1453] [,1454] [,1455] [,1456] [,1457]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 5 5 5 5 5 5 5 5 6
## [4,] 8 8 8 9 9 9 9 10 7
## [5,] 10 10 11 10 10 10 11 11 8
## [6,] 11 12 12 11 11 12 12 12 9
## [7,] 13 13 13 12 13 13 13 13 10
## [,1458] [,1459] [,1460] [,1461] [,1462] [,1463] [,1464] [,1465] [,1466]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 8 8 8 8
## [6,] 9 9 9 10 10 10 11 11 12
## [7,] 11 12 13 11 12 13 12 13 13
## [,1467] [,1468] [,1469] [,1470] [,1471] [,1472] [,1473] [,1474] [,1475]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 9 9 9 9 9 9 10 10 10
## [6,] 10 10 10 11 11 12 11 11 12
## [7,] 11 12 13 12 13 13 12 13 13
## [,1476] [,1477] [,1478] [,1479] [,1480] [,1481] [,1482] [,1483] [,1484]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 8 8 8 8 8 8 8 8
## [5,] 11 9 9 9 9 9 9 10 10
## [6,] 12 10 10 10 11 11 12 11 11
## [7,] 13 11 12 13 12 13 13 12 13
## [,1485] [,1486] [,1487] [,1488] [,1489] [,1490] [,1491] [,1492] [,1493]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 6 6 6 6 6 6 6 7 7
## [4,] 8 8 9 9 9 9 10 8 8
## [5,] 10 11 10 10 10 11 11 9 9
## [6,] 12 12 11 11 12 12 12 10 10
## [7,] 13 13 12 13 13 13 13 11 12
## [,1494] [,1495] [,1496] [,1497] [,1498] [,1499] [,1500] [,1501] [,1502]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 8 9
## [5,] 9 9 9 9 10 10 10 11 10
## [6,] 10 11 11 12 11 11 12 12 11
## [7,] 13 12 13 13 12 13 13 13 12
## [,1503] [,1504] [,1505] [,1506] [,1507] [,1508] [,1509] [,1510] [,1511]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 4 4 4 4 4 4 4 4
## [3,] 7 7 7 7 8 8 8 8 8
## [4,] 9 9 9 10 9 9 9 9 10
## [5,] 10 10 11 11 10 10 10 11 11
## [6,] 11 12 12 12 11 11 12 12 12
## [7,] 13 13 13 13 12 13 13 13 13
## [,1512] [,1513] [,1514] [,1515] [,1516] [,1517] [,1518] [,1519] [,1520]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 4 5 5 5 5 5 5 5 5
## [3,] 9 6 6 6 6 6 6 6 6
## [4,] 10 7 7 7 7 7 7 7 7
## [5,] 11 8 8 8 8 8 8 8 8
## [6,] 12 9 9 9 9 10 10 10 11
## [7,] 13 10 11 12 13 11 12 13 12
## [,1521] [,1522] [,1523] [,1524] [,1525] [,1526] [,1527] [,1528] [,1529]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 9 9 9 9 9 9 10
## [6,] 11 12 10 10 10 11 11 12 11
## [7,] 13 13 11 12 13 12 13 13 12
## [,1530] [,1531] [,1532] [,1533] [,1534] [,1535] [,1536] [,1537] [,1538]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 8 8 8 8 8 8
## [5,] 10 10 11 9 9 9 9 9 9
## [6,] 11 12 12 10 10 10 11 11 12
## [7,] 13 13 13 11 12 13 12 13 13
## [,1539] [,1540] [,1541] [,1542] [,1543] [,1544] [,1545] [,1546] [,1547]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 9 9 9 9 10
## [5,] 10 10 10 11 10 10 10 11 11
## [6,] 11 11 12 12 11 11 12 12 12
## [7,] 12 13 13 13 12 13 13 13 13
## [,1548] [,1549] [,1550] [,1551] [,1552] [,1553] [,1554] [,1555] [,1556]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 8 8
## [5,] 9 9 9 9 9 9 10 10 10
## [6,] 10 10 10 11 11 12 11 11 12
## [7,] 11 12 13 12 13 13 12 13 13
## [,1557] [,1558] [,1559] [,1560] [,1561] [,1562] [,1563] [,1564] [,1565]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 7 7 7 8 8 8
## [4,] 8 9 9 9 9 10 9 9 9
## [5,] 11 10 10 10 11 11 10 10 10
## [6,] 12 11 11 12 12 12 11 11 12
## [7,] 13 12 13 13 13 13 12 13 13
## [,1566] [,1567] [,1568] [,1569] [,1570] [,1571] [,1572] [,1573] [,1574]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 5 5 5 6 6 6 6 6 6
## [3,] 8 8 9 7 7 7 7 7 7
## [4,] 9 10 10 8 8 8 8 8 8
## [5,] 11 11 11 9 9 9 9 9 9
## [6,] 12 12 12 10 10 10 11 11 12
## [7,] 13 13 13 11 12 13 12 13 13
## [,1575] [,1576] [,1577] [,1578] [,1579] [,1580] [,1581] [,1582] [,1583]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 9 9 9 9 10
## [5,] 10 10 10 11 10 10 10 11 11
## [6,] 11 11 12 12 11 11 12 12 12
## [7,] 12 13 13 13 12 13 13 13 13
## [,1584] [,1585] [,1586] [,1587] [,1588] [,1589] [,1590] [,1591] [,1592]
## [1,] 3 3 3 3 3 3 3 3 3
## [2,] 6 6 6 6 6 6 7 7 7
## [3,] 8 8 8 8 8 9 8 8 8
## [4,] 9 9 9 9 10 10 9 9 9
## [5,] 10 10 10 11 11 11 10 10 10
## [6,] 11 11 12 12 12 12 11 11 12
## [7,] 12 13 13 13 13 13 12 13 13
## [,1593] [,1594] [,1595] [,1596] [,1597] [,1598] [,1599] [,1600] [,1601]
## [1,] 3 3 3 3 4 4 4 4 4
## [2,] 7 7 7 8 5 5 5 5 5
## [3,] 8 8 9 9 6 6 6 6 6
## [4,] 9 10 10 10 7 7 7 7 7
## [5,] 11 11 11 11 8 8 8 8 8
## [6,] 12 12 12 12 9 9 9 9 10
## [7,] 13 13 13 13 10 11 12 13 11
## [,1602] [,1603] [,1604] [,1605] [,1606] [,1607] [,1608] [,1609] [,1610]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 7 7 7
## [5,] 8 8 8 8 8 9 9 9 9
## [6,] 10 10 11 11 12 10 10 10 11
## [7,] 12 13 12 13 13 11 12 13 12
## [,1611] [,1612] [,1613] [,1614] [,1615] [,1616] [,1617] [,1618] [,1619]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 7 7 7 7 7 7 8 8 8
## [5,] 9 9 10 10 10 11 9 9 9
## [6,] 11 12 11 11 12 12 10 10 10
## [7,] 13 13 12 13 13 13 11 12 13
## [,1620] [,1621] [,1622] [,1623] [,1624] [,1625] [,1626] [,1627] [,1628]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 6 6 6 6 6 6
## [4,] 8 8 8 8 8 8 8 9 9
## [5,] 9 9 9 10 10 10 11 10 10
## [6,] 11 11 12 11 11 12 12 11 11
## [7,] 12 13 13 12 13 13 13 12 13
## [,1629] [,1630] [,1631] [,1632] [,1633] [,1634] [,1635] [,1636] [,1637]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 6 6 6 7 7 7 7 7 7
## [4,] 9 9 10 8 8 8 8 8 8
## [5,] 10 11 11 9 9 9 9 9 9
## [6,] 12 12 12 10 10 10 11 11 12
## [7,] 13 13 13 11 12 13 12 13 13
## [,1638] [,1639] [,1640] [,1641] [,1642] [,1643] [,1644] [,1645] [,1646]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 5 5 5
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 9 9 9 9 10
## [5,] 10 10 10 11 10 10 10 11 11
## [6,] 11 11 12 12 11 11 12 12 12
## [7,] 12 13 13 13 12 13 13 13 13
## [,1647] [,1648] [,1649] [,1650] [,1651] [,1652] [,1653] [,1654] [,1655]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 5 5 5 5 5 5 6 6 6
## [3,] 8 8 8 8 8 9 7 7 7
## [4,] 9 9 9 9 10 10 8 8 8
## [5,] 10 10 10 11 11 11 9 9 9
## [6,] 11 11 12 12 12 12 10 10 10
## [7,] 12 13 13 13 13 13 11 12 13
## [,1656] [,1657] [,1658] [,1659] [,1660] [,1661] [,1662] [,1663] [,1664]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 9 9
## [5,] 9 9 9 10 10 10 11 10 10
## [6,] 11 11 12 11 11 12 12 11 11
## [7,] 12 13 13 12 13 13 13 12 13
## [,1665] [,1666] [,1667] [,1668] [,1669] [,1670] [,1671] [,1672] [,1673]
## [1,] 4 4 4 4 4 4 4 4 4
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 8 8 8 8 8 9
## [4,] 9 9 10 9 9 9 9 10 10
## [5,] 10 11 11 10 10 10 11 11 11
## [6,] 12 12 12 11 11 12 12 12 12
## [7,] 13 13 13 12 13 13 13 13 13
## [,1674] [,1675] [,1676] [,1677] [,1678] [,1679] [,1680] [,1681] [,1682]
## [1,] 4 4 4 4 4 4 4 5 5
## [2,] 7 7 7 7 7 7 8 6 6
## [3,] 8 8 8 8 8 9 9 7 7
## [4,] 9 9 9 9 10 10 10 8 8
## [5,] 10 10 10 11 11 11 11 9 9
## [6,] 11 11 12 12 12 12 12 10 10
## [7,] 12 13 13 13 13 13 13 11 12
## [,1683] [,1684] [,1685] [,1686] [,1687] [,1688] [,1689] [,1690] [,1691]
## [1,] 5 5 5 5 5 5 5 5 5
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 7 7 7 7 7
## [4,] 8 8 8 8 8 8 8 8 9
## [5,] 9 9 9 9 10 10 10 11 10
## [6,] 10 11 11 12 11 11 12 12 11
## [7,] 13 12 13 13 12 13 13 13 12
## [,1692] [,1693] [,1694] [,1695] [,1696] [,1697] [,1698] [,1699] [,1700]
## [1,] 5 5 5 5 5 5 5 5 5
## [2,] 6 6 6 6 6 6 6 6 6
## [3,] 7 7 7 7 8 8 8 8 8
## [4,] 9 9 9 10 9 9 9 9 10
## [5,] 10 10 11 11 10 10 10 11 11
## [6,] 11 12 12 12 11 11 12 12 12
## [7,] 13 13 13 13 12 13 13 13 13
## [,1701] [,1702] [,1703] [,1704] [,1705] [,1706] [,1707] [,1708] [,1709]
## [1,] 5 5 5 5 5 5 5 5 6
## [2,] 6 7 7 7 7 7 7 8 7
## [3,] 9 8 8 8 8 8 9 9 8
## [4,] 10 9 9 9 9 10 10 10 9
## [5,] 11 10 10 10 11 11 11 11 10
## [6,] 12 11 11 12 12 12 12 12 11
## [7,] 13 12 13 13 13 13 13 13 12
## [,1710] [,1711] [,1712] [,1713] [,1714] [,1715] [,1716]
## [1,] 6 6 6 6 6 6 7
## [2,] 7 7 7 7 7 8 8
## [3,] 8 8 8 8 9 9 9
## [4,] 9 9 9 10 10 10 10
## [5,] 10 10 11 11 11 11 11
## [6,] 11 12 12 12 12 12 12
## [7,] 13 13 13 13 13 13 13In how many ways can the letters in the word 'Population' be arranged?10!/(2!*2!) = 907200
22. Consider the following data:
x 5 6 7 8 9 p(x) 0.1 0.2 0.3 0.2 0.2
x <- c(5, 6, 7, 8, 9)
px<- c(.1, .2, .3, .2, .2)
datatable <- data.frame(x,px)
print(datatable)## x px
## 1 5 0.1
## 2 6 0.2
## 3 7 0.3
## 4 8 0.2
## 5 9 0.2Step 1. Find the expected value E( X ). Round your answer to one decimal place.
expectedvalue <- sum(datatable$x * datatable$px)
expectedvalue## [1] 7.2Step 2. Find the variance. Round your answer to one decimal place.
variance <- sum((datatable$x - expectedvalue)^2 * datatable$px)
variance## [1] 1.56Step 3. Find the standard deviation. Round your answer to one decimal place.
standard_deviation <- sqrt(variance)
standard_deviation## [1] 1.249Step 4. Find the value of P(X ³ 9). Round your answer to one decimal place.
.2
Step 5. Find the value of P(X £ 7). Round your answer to one decimal place.
.6
Suppose a basketball player has made 188 out of 376 free throws. If the player makes the next 3 free throws, I will pay you $23. Otherwise you pay me $4.Step 1. Find the expected value of the proposition. Round your answer to two decimal places. 188/376=.5 [(.5.55)*23]-(.875*4)= $-.62
Step 2. If you played this game 994 times how much would you expect to win or lose? (Losses must be entered as negative.)
-.62*994= -616.28
Flip a coin 11 times. If you get 8 tails or less, I will pay you $1. Otherwise you pay me $7.Step 1. Find the expected value of the proposition. Round your answer to two decimal places.
pw <- pbinom(8, size = 11, prob = .5)
expected_value <- round(1 * pw - 7 * (1-pw), 2)
expected_value## [1] 0.74Step 2. If you played this game 615 times how much would you expect to win or lose? (Losses must be entered as negative.)
expected_value*615## [1] 455.1If you draw two clubs on two consecutive draws from a standard deck of cards you win $583. Otherwise you pay me $35. (Cards drawn without replacement.)Step 1. Find the expected value of the proposition. Round your answer to two decimal places.
pw <- (13/52)*(12/51)
exp_value <- round(pw * 583 - (1 - pw) * 35, 2)
exp_value## [1] 1.35Step 2. If you played this game 632 times how much would you expect to win or lose? (Losses must be entered as negative.)
exp_value*632## [1] 853.2A quality control inspector has drawn a sample of 10 light bulbs from a recent production lot. If the number of defective bulbs is 2 or less, the lot passes inspection. Suppose 30% of the bulbs in the lot are defective. What is the probability that the lot will pass inspection? (Round your answer to 3 decimal places)pass <- pbinom(2, 10, .3)
print(round(pass,3))## [1] 0.383A quality control inspector has drawn a sample of 5 light bulbs from a recent production lot. Suppose that 30% of the bulbs in the lot are defective. What is the expected value of the number of defective bulbs in the sample? Do not round your answer. exp_value <- 5*.3
exp_value## [1] 1.5The auto parts department of an automotive dealership sends out a mean of 5.5 special orders daily. What is the probability that, for any day, the number of special orders sent out will be more than 5? (Round your answer to 4 decimal places)round(ppois(5, 5.5, lower=FALSE), 4)## [1] 0.4711At the Fidelity Credit Union, a mean of 5.7 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 4 customers will arrive? (Round your answer to 4 decimal places)round(ppois(4, 5.7, lower=FALSE), 4)## [1] 0.6728The computer that controls a bank's automatic teller machine crashes a mean of 0.4 times per day. What is the probability that, in any 7-day week, the computer will crash no more than 1 time? (Round your answer to 4 decimal places)crash <- .4*7
round(ppois(1, crash), 4)## [1] 0.2311A town recently dismissed 8 employees in order to meet their new budget reductions. The town had 6 employees over 50 years of age and 19 under 50. If the dismissed employees were selected at random, what is the probability that more than 1 employee was over 50? Write your answer as a fraction or a decimal number rounded to three decimal places.round(phyper(1, m=6, n=19, 8, lower.tail=FALSE), 3)## [1] 0.651Unknown to a medical researcher, 10 out of 25 patients have a heart problem that will result in death if they receive the test drug. Eight patients are randomly selected to receive the drug and the rest receive a placebo. What is the probability that less than 7 patients will die? Write your answer as a fraction or a decimal number rounded to three decimal places.round(phyper(6, m=10, n=15, 8), 3)## [1] 0.998The weights of steers in a herd are distributed normally. The variance is 40,000 and the mean steer weight is 1300 lbs. Find the probability that the weight of a randomly selected steer is greater than 979 lbs. (Round your answer to 4 decimal places)round(pnorm(979, 1300, 200, lower.tail = FALSE), 4)## [1] 0.9458SVGA monitors manufactured by TSI Electronics have life spans that have a normal distribution with a variance of 1,960,000 and a mean life span of 11,000 hours. If a SVGA monitor is selected at random, find the probability that the life span of the monitor will be more than 8340 hours. (Round your answer to 4 decimal places)round(pnorm(8340, 11000, sqrt(1960000), lower.tail = FALSE), 4)## [1] 0.9713Suppose the mean income of firms in the industry for a year is 80 million dollars with a standard deviation of 3 million dollars. If incomes for the industry are distributed normally, what is the probability that a randomly selected firm will earn between 83 and 85 million dollars? (Round your answer to 4 decimal places)round((pnorm(83, 80, 3, lower.tail = FALSE)-pnorm(85, 80, 3, lower.tail = FALSE)), 4)## [1] 0.1109Suppose GRE Verbal scores are normally distributed with a mean of 456 and a standard deviation of 123. A university plans to offer tutoring jobs to students whose scores are in the top 14%. What is the minimum score required for the job offer? Round your answer to the nearest whole number, if necessary.round (qnorm(.14, 456, 123, lower.tail = FALSE),0)## [1] 589The lengths of nails produced in a factory are normally distributed with a mean of 6.13 centimeters and a standard deviation of 0.06 centimeters. Find the two lengths that separate the top 7% and the bottom 7%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.round(qnorm(.07, 6.13, .06), 3)## [1] 6.041round(qnorm(.93, 6.13, .06), 3)## [1] 6.219An English professor assigns letter grades on a test according to the following scheme.M <- 78.8
SD <- 9.8
x1 <- .55
x2 <- .20
round(qnorm(.55, 78.8, 9.8),0)## [1] 80round(qnorm(.2, 78.8, 9.8),0)## [1] 71Suppose ACT Composite scores are normally distributed with a mean of 21.2 and a standard deviation of 5.4. A university plans to admit students whose scores are in the top 45%. What is the minimum score required for admission? Round your answer to the nearest tenth, if necessary.round(qnorm(.45, mean = 21.2, sd = 5.4, lower.tail = FALSE), 0)## [1] 22
Consider the probability that less than 11 out of 151 students will not graduate on time. Assume the probability that a given student will not graduate on time is 9%. Approximate the probability using the normal distribution. (Round your answer to 4 decimal places.)round(pbinom(10, 151, .09), 4)## [1] 0.192The mean lifetime of a tire is 48 months with a standard deviation of 7. If 147 tires are sampled, what is the probability that the mean of the sample would be greater than 48.83 months? (Round your answer to 4 decimal places)round(pnorm(48.83, mean=48, sd=(7/sqrt(147)), lower.tail = FALSE),4)## [1] 0.0753The quality control manager at a computer manufacturing company believes that the mean life of a computer is 91 months, with a standard deviation of 10. If he is correct, what is the probability that the mean of a sample of 68 computers would be greater than 93.54 months? (Round your answer to 4 decimal places)round(pnorm(93.54, mean=91, sd=(10/sqrt(68)), lower.tail = FALSE),4)## [1] 0.0181A director of reservations believes that 7% of the ticketed passengers are no-shows. If the director is right, what is the probability that the proportion of no-shows in a sample of 540 ticketed passengers would differ from the population proportion by less than 3%? (Round your answer to 4 decimal places)A bottle maker believes that 23% of his bottles are defective. If the bottle maker is accurate, what is the probability that the proportion of defective bottles in a sample of 602 bottles would differ from the population proportion by greater than 4%? (Round your answer to 4 decimal places)se <- sqrt(.23 * (1-.23)/602)
upper <- pnorm(.27, mean = .23, sd = se, lower.tail=TRUE)
lower <- pnorm(.19, mean = .23, sd = se, lower.tail=TRUE)
diff <- upper - lower
print(round(diff,4))## [1] 0.9803A research company desires to know the mean consumption of beef per week among males over age 48. Suppose a sample of size 208 is drawn with x ̅ = 3.9. Assume ® = 0.8 . Construct the 80% confidence interval for the mean number of lb. of beef per week among males over 48. (Round your answers to 1 decimal place) n <- 208
m <- 3.9
sd <- .8
se <- .8/sqrt(208)
t <- qt((1 - .8)/2, 207)
c1 <- round((m + t * se),1)
c2 <- round((m - t * se),1)
print(paste0("lower",c1))## [1] "lower3.8"print(paste0("upper",c2))## [1] "upper4"An economist wants to estimate the mean per capita income (in thousands of dollars) in a major city in California. Suppose a sample of size 7472 is drawn with x ̅ = 16.6. Assume ® = 11 . Construct the 98% confidence interval for the mean per capita income. (Round your answers to 1 decimal place) n <- 7472
m <- 16.6
sd <- 11
se <- sd/sqrt(n)
t <- qt((1 - .98) / 2, n- 1)
c1 <- round((m + t * se),1)
c2 <- round((m - t * se),1)
print(paste0("lower",c1))## [1] "lower16.3"print(paste0("upper",c2))## [1] "upper16.9"47. Find the value of t such that 0.05 of the area under the curve is to the left of t. Assume the degrees of freedom equals 26.
Step 1. Choose the picture which best describes the problem.
the second one
Step 2. Write your answer below.
print(qt(0.05,df=26))## [1] -1.705618The following measurements ( in picocuries per liter ) were recorded by a set of helium gas detectors installed in a laboratory facility:
383.6, 347.1, 371.9, 347.6, 325.8, 337Step 1. Calculate the sample mean for the given sample data. (Round answer to 2 decimal places)
gas <- c(383.6, 347.1, 371.9, 347.6, 325.8, 337)
m <- sum(gas)/6
print(round(m,2))## [1] 352.17Step 2. Calculate the sample standard deviation for the given sample data. (Round answer to 2 decimal places)
sd <- sqrt(var(gas))
print(round(sd,2))## [1] 21.68Step 3. Find the critical value that should be used in constructing the confidence interval. (Round answer to 3 decimal places)
n = length(gas)
cv <- qt(.10/2, (n - 1), lower.tail = FALSE)
print(round(cv,3))## [1] 2.015Step 4. Construct the 90% confidence interval. (Round answer to 2 decimal places)
Upper <- m + cv * sd / sqrt(6)
Lower <- m - cv * sd / sqrt(6)
print(paste0("Upper ", round(Upper,2)))## [1] "Upper 370"print(paste0("Lower ", round(Lower,2)))## [1] "Lower 334.34"
A random sample of 16 fields of spring wheat has a mean yield of 46.4 bushels per acre and standard deviation of 2.45 bushels per acre. Determine the 80% confidence interval for the true mean yield. Assume the population is normally distributed. Step 1. Find the critical value that should be used in constructing the confidence interval. (Round answer to 3 decimal places)
n <-16
m <-46.4
sd <-2.45
cv <-qt((1-.8)/2,n-1,lower.tail = FALSE)
print(round(cv, 3))## [1] 1.341Step 2. Construct the 80% confidence interval. (Round answer to 1 decimal place)
Upper <- m + cv * sd / sqrt(16)
lower <- m - cv * sd / sqrt(16)
print(paste0("Upper ", round(Upper,1)))## [1] "Upper 47.2"print(paste0("lower ", round(lower,1)))## [1] "lower 45.6"A toy manufacturer wants to know how many new toys children buy each year. She thinks the mean is 8 toys per year. Assume a previous study found the standard deviation to be 1.9. How large of a sample would be required in order to estimate the mean number of toys bought per child at the 99% confidence level with an error of at most 0.13 toys? (Round your answer up to the next integer)zvalue = qnorm(1-(1-.99)/2)
n50 = ceiling(((zvalue*1.9)/0.13)^2)
print(n50)## [1] 1418A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. He believes that the mean is 12.6. Assume the variance is known to be 3.61. How large of a sample would be required in order to estimate the mean number of reproductions per hour at the 95% confidence level with an error of at most 0.19 reproductions? (Round your answer up to the next integer)zv = qnorm(1-(1-.95)/2)
n = ceiling(((zv*1.9)/0.19)^2)
print(n)## [1] 385The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level.Step 1. Suppose a sample of 2089 tenth graders is drawn. Of the students sampled, 1734 read above the eighth grade level. Using the data, estimate the proportion of tenth graders reading at or below the eighth grade level. (Write your answer as a fraction or a decimal number rounded to 3 decimal places)
n <- 2089
Above <- 1734
Below <- n - Above
x <- Below/n
round(x, 3)## [1] 0.17Step 2. Suppose a sample of 2089 tenth graders is drawn. Of the students sampled, 1734 read above the eighth grade level. Using the data, construct the 98% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level. (Round your answers to 3 decimal places)
SE <- sqrt((x)*(1-x)/n)
z <- 2.326
CI1 <- round(x - z * SE, 3)
CI2 <- round( x + z * SE, 3)
round(CI1, 3)## [1] 0.151round(CI2, 3)## [1] 0.189round(CI1*n, 3)## [1] 315.439round(CI2*n, 3)## [1] 394.821An environmentalist wants to find out the fraction of oil tankers that have spills each month.Step 1. Suppose a sample of 474 tankers is drawn. Of these ships, 156 had spills. Using the data, estimate the proportion of oil tankers that had spills. (Write your answer as a fraction or a decimal number rounded to 3 decimal places)
n53 <- 474
spills <- 156
P <- round(spills/n53, 3)
P## [1] 0.329[Continued on next page ….]
Step 2. Suppose a sample of 474 tankers is drawn. Of these ships, 156 had spills. Using the data, construct the 95% confidence interval for the population proportion of oil tankers that have spills each month. (Round your answers to 3 decimal places)
SD <- sqrt ((P)*(1-P)/474)
round(qnorm((1-(1-0.95)/2),P,SD, lower.tail = FALSE),3)## [1] 0.287round(qnorm((1-(1-0.95)/2),P,SD),3)## [1] 0.371