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Instructions: Use the space provided to enter your answers. Be sure to use the Microsoft Equation Editor to show answers and calculations. ๏ท 1) Suppose x is a random variable best described by a uniform probability distribution with c=10 and d=30 Show calculations for each computation. ๏ท a) Find f(x) ๏ท 1 { b) Find the mean and standard deviation. 20 , 10 < ๐ฅ < 30 0 ๐๐กโ๐๐๐ค๐๐ ๐ 10 Mean =20, sd = โ3 If x has a uniform distribution on the interval (c,d) Then mean = ๐+๐ 2 ๐โ๐ Sd = โ12 If c=10 and d=30 Then mean = ๐โ๐ Sd = โ12 = ๐+๐ 2 30โ10 โ12 = 10+30 2 20 = 2โ3 = = 40 2 = 20 10 โ3 ๏ท 2) Find the value of of the standard normal random variable for the following probabilities. Show calculations for each computation. ๏ท a) P(z โค z_0 )=.95 z0 = 1.645 ๏ท b) P(-2 < z < z_0)=.9710 z0=2.512 a) Using normal table P(zโค1.645)=0.95 b) P(-2 < z < z0) =P(z<z0)-P(z<-2)=P(z<z0)-0.023=0.9710 P(z<z0)=0.9710+0.023=0.994 z0=2.512 (using normal table) ๏ท 3) Suppose is a normally distributed random variable with ฮผ=11and ฯ=2. Find each of the following. Show calculations for each computation. ๏ท a) P(10โคxโค12) 0.383 ๏ท b) P(xโฅ13.24) 0.131 Z=(X- ฮผ)/ ฯ =(X-11)/2has a standard normal distribution a) P(10โคxโค12)=P((10-11)/2โคZโค(12-11)/2)= P(-1/2โคZโค1/2)=P(zโค0.5)- P(zโค0.5)=0.383 b) P(xโฅ13.24)=P(Zโฅ(13.24-11)/2)=P(Zโฅ1.12)=1-P(Zโค1.12)= 1-0.869=0.131 ๏ท 4) Refer to the Chance (Winter 2001) study of students who paid a private tutor to help them improve their SAT scores, presented in Exercise 2.101 (p.74). The table summarizing the changes in both the SAT-Mathematics and SAT-Verbal scores for these students is reproduced here. Assume that both distributions of SAT score changes are approximately normal. SAT-Math SAT-Verbal 19 7 Mean change in score 65 49 Standard deviation of changes in score. Compute the probability that a student increases his or her score on the SAT-Math test by at least 50 points? Show calculations. X= increase of the score X has a normal distribution with Mean = 19 and sd=65 Z= (X-19)/65 has a standard normal distribution P(X>50)=P(Z>(50-19)/65)=P(Z>0.477)=0.317 Answer: 0.317