Question

Suppose that the weight of Florida navel oranges is normally distributed with mean µ = 8 ounces, and standard deviation σ = 1.5 ounces. (a) (1 point) State the model in notation form. (b) (2 points) What proportion of oranges weigh more than 11.5 ounces? (c) (2 points) What proportion of oranges weigh less than 8.7 ounces? (d) (2 points) What proportion of oranges weigh between 6.2 and 7 ounces? Page 3 (e) (5 points) What are the median, mode, first quartile, third quartile and IQR for the oranges? (f) (2 points) Find the 80th percentile of the distribution of X. (g) Suppose another strain of navel oranges from California has a mean µ = 8.5 ounces, and standard deviation σ = 1.7 ounces. A Florida orange is randomly selected and weighs 12 ounces, while a California orange is randomly selected and weighs 13 ounces. Which one strain of orange is ”relatively” bigger for its average size?

Answer 1

Given that M-8,0-1.15 a) Xe N( 8 1.5) b) P(x211.5) - 1-P (X= 11.5) = 1-PC Videos 11.58 1.5 1-PC2% 2.333) 21-0.9902 -0.0098 Poeportion et Oranges weight more than 11.5 ounces is 0.0098 c) P(X8.9) = P(X-< 8-168 p(250-467) - 0.6798 Poopootfors of cranges weight less then 84 ounces P& 0.6498 d) P(6.25 X47) - P(X47)-P6X56.2) pero siem) - BOX 36.27) - Plzs -0.6+) -P (254.2) = 0.2514 -0.115) = 0.1363 Pooportion orarges weight between 6.2 and 4 ounces is 0.1363

e) Median, Mean and Mode are equal for Normal Sandom Variable :: Median = 8 Modez8 QI => P(X+QI) -0.25 PC key < 0.53 )=0-25 Plzs 20-25)-0.25 Plzs -0.6745)=0.25 :-0-6745 - 8,-8 -> 4.-8-(0.674561.5) 1.5 11-6.9883 Q3 POULQ3) -0.15 P(xus 93-8 )-0.75 1.5 Plzs 20.75)-075 PIZZ 0.6745)=0.75 0.6745 - 938 - 03-8+(0.6745*1.5) 1.5 Q3-9.0118 TeR = Q3-Q, = 9.0118-6.9883 > 2.0235 TzR 2 2.0235

Mr. Average weight of Californial naved oranges 80 percontic -e Po.80 P(XSP080)-0.80 P( 1-4The8 ) =0-80 plzs 20.80)-0.80 PC ZS 0-8416)-0.80 >> Po.807 81(0.8416*1.5) -9.9624 percentile 9 & 9-2624 9 We have Mli: Average weight of Florida novel Oranges 0.8416 = P0-80-8 1.5 Soth We have l128, T1-5, n =12 U2=8.5, 82=117 N2=13 1.5 -0.433 Standard erooo for els is ng standard error for ez is a Vni 20.471 v 13 Since Standard foror for asconge. weight of california is higher than for florida con Cordlude that ploda "Oanges are bigger in Size compare to California oranges, So we

RR Console > #b) > pnorm (2.333) [1] 0.9901759 > #c) > pnorm (0.467) [1] 0.6797501 > #d) > pnorm (-0.67) [1] 0.2514289 [1] 0.1150697 > qnoxm (0.5,8,1.5) [1] > groum (0.25) [1] -0.6744898 > oroit. 10.75) [1] 0.6744898 > qnorm (0.80) [1] 0.8416212 2