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# hw4 - STAT 350 Spring 2009 Homework#4 Solution covers...

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STAT 350 – Spring 2009 Homework #4 Solution covers through Lecture G 1. X and Y are independent random variables () ~ Normal 30, 15 ~ Normal 20, 10 xx yy X Y μσ =− = == Let W = 3 Y – 5 X +100 a. Find E( W ) E(3Y – 5X +100) = E(3Y) + E(-5X) + E(100) = 3E(Y) + (-5)E(X) + 100 = 3(20) + (-5)(-30) + 100 =310 b. Find SD( W ) Var(3Y – 5X +100) = Var(3Y) + Var(-5X) + Var(100) = 3 2 Var(Y) + (-5) 2 Var(X) + 0 = 9(10 2 ) + 25(15 2 ) =6525 SD(W) = sqrt(6525) = 80.77747211 c. Find P( W > 250) ( ) ( ) 310 250 310 250 0.74278 6525 6525 1 0.74278 1 0.74 1 0.74 1 0.2296 W PW P PZ −− ⎛⎞ >= > = > ⎜⎟ ⎝⎠ ≤− ≈− =−Φ− = 0.7704 d. Find P(300 < W < 500) ( ) ( ) 300 240 240 500 240 0.74278 3.21872 6525 6525 6525 0.74 3.22 3.22 0.74 0.9994 0.7704 W PP =< < = < < ≈< < = Φ Φ= Z =0.2290 Homework #4 - Solution Page 1 of 1

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2. At Farmer Jack's Pumpkin Patch, the weight (in ounces) of ripe pumpkins is normally distributed with a mean of 240 ounces and a standard deviation of 50 ounces. For the following problems, let X 1 = the weight (in ounces) of a single randomly selected pumpkin from this patch X 1 ~ Normal( μ = 240, σ = 50) a. What is the probability that a randomly selected pumpkin from Farmer Jack's Pumpkin Patch weighs less than 100 ounces? () ( ) 1 1 240 100 240 100 2.8 2.8 0.0026 50 50 X PX P PZ −− ⎛⎞ <= < = < = Φ = ⎜⎟ ⎝⎠ b. What is the probability that a randomly selected pumpkin from Farmer Jack's Pumpkin Patch weighs less than 302 ounces? ( ) 1 1 240 302 240 302 1.24 1.24 0.8925 50 50 X P < = < = Φ = c. What is the probability that a randomly selected pumpkin from Farmer Jack's Pumpkin Patch weighs more than 220 ounces? ( ) 1 1 240 220 240 220 0.40 50 50 1 0.40 1 0.40 1 0.3446 X P >= > = > =− ≤− =−Φ− = 0.6554 d. What is the probability that a randomly selected pumpkin from Farmer Jack's Pumpkin Patch weighs more than 359 ounces? ( ) 1 1 240 359 240 359 2.38 50 50 1 2.38 1 2.38 1 0.9913 X P > = > =−Φ = 0.0087 e. What is the probability that a randomly selected pumpkin from Farmer Jack's Pumpkin Patch weighs between 302 and 359 ounces? ()() 1 1 240 302 240 359 240 302 359 1.24 2.38 50 50 50 2.38 1.24 0.9913 0.8925 X P << = < < = < < −Φ = = 0.0988 Homework #4 - Solution Page 2 of 2
f. Find the 80th percentile of weights for this population of pumpkins. We want to find c , such that P( X 1 c ) = 0.80 We must first find z , such that P( Z z ) = 0.80 So find z , such that Φ ( z ) = 0.80 From the standard normal table, we find: Φ (0.84) = 0.7995 and Φ (0.85) = 0.8023 z is closest to 0.84. ( ) () () () 1 This is ~Normal(240,50) 282 just like 1 0.80 0.84 50 0.84 50 50 240 0.84 50 240 0.80 282 X PZ PX = =≤ ⎛⎞ ⎜⎟ =+ + ⎝⎠ ⇒= ±²³²´ ±²²³ ² ²´ The 80th percentile is 282. g. Find the 20th percentile of weights for this population of pumpkins. We want to find c , such that P( X 1 c ) = 0.20 We must first find z , such that P( Z z ) = 0.20 So find z , such that Φ ( z ) = 0.20 From the standard normal table, we find: Φ (-0.84) = 0.2005 and Φ (-0.85) = 0.1977 z is closest to -0.84. 1 This is ~Normal(240,50) 198 just like 1 0.20 0.84 50 0.84 50 50 240 0.84 50 240 0.20 198 X = + ±²²³²²´ The 20th percentile is 198. h. How heavy (in ounces) are the middle 90% of Farmer Jack's pumpkins? (That is, find the 5th and 95th percentiles of pumpkin weights.) The 5th and 95th percentiles of the standard normal distribution are -1.645 and +1.645.

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hw4 - STAT 350 Spring 2009 Homework#4 Solution covers...

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