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#5.48#5.48 ACT scores of high school seniors. The scores of high school seniors on the ACT college entrance examination in 2003 had mean =20.8 and SD =4.8. The distribution of scores is only roughly Normal.(a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher?(b) Now taken as SRS of 25 students who took the test. What are the mean and standard deviation of the sample means score x of these 25 students?(c) What is the approximate probability that the mean scoreof these students is 23 or higher?(d) Which of your two Normal probability calculations in (a) and (c) is more accurate? Why?Solutions:Given that, =20.8 and SD =4.8(a)Required Prob: P( X >= 23 )P( X >= 23 ) = 1 - P( X < 23 )P( X < 23 ) =0.6766(by using excel normdist() function, click on the respective value for how it is calculated)P( X >= 23 ) = 1 - P( X < 23 ) = 1-0.6766 =0.3234(b)For this problem lets recollect thatBy the properties of means and variances of random variables, the mean and variance of the sample mean are the following:Note: Here x-bar also refer as MNow, when n=25Expected value of M = 20.8Standard deviation of M = / sqrt(n) = 4.8 / sqrt(25) = 4.8 / 5 = 0.96( c)Required Prob: P( sample mean(M) >= 23 )P( M >= 23 ) = 1 - P( M < 23 )P( M < 23 ) =0.9890(by using eMcel normdist() function, click on the respective value for how it is calculated)P( M >= 23 ) = 1 - P( M < 23 ) = 1-0.9890 =0.0110(d)Normal probability calculation in (c) is more accurate, because it is capturing the current sample's variability,whereas (a) captures overall population's variability

#5.52#5.52 A Lottery payoff. A $1 bet in a state lotterys Pick 3 game pays $500 if the three-digit number you choose exactly matchesthe winning number, which is drawn at random. Here is the distribution of the payoff X:Payoff X$0$500Probability0.9990.001Each days drawing is independent of other drawings.(a) What are the mean and SD of x?(b) Joe buys a Pick 3 ticket twice a week. What does the law of large numbers say about the average payoff Joe receives from his bets?(c) What does the central limit theorem say about the distribution of Joes average payoff after 104 bets in a year?(d) Joe comes out ahead for the year if his average payoff is greater than $1 (the amount he spent each day on a ticket).What is the probability that Joe ends the year ahead?Solutions:(a)x$0$500Imp note: Generally there should be loss factor, here it should be -1 in place of 0P(x)0.9990.001but considering as it is to avoide the confusionmean = xp(x) = 0*0.999 + 500*0.001 =0.5SD = sqrt ( V(x) )V(x) = x*x*p(x) - (mean*mean) = 0*0*0.999 + 500*500*0.001 - 0.5*0.5 =249.75SD = sqrt ( 249.75 ) =15.803(b)From the law of large numbers, the average payoff joe receives from his bets will be close population's average payoff(c)The central limit theorem (CLT) states conditions under which the mean of a sufficientlylarge number of independent random variables, each with finite mean and variance, will be approximately normally distributed(d)Required prob: P( X > 1)As here, n = 104Mean of sample mean = 0.5SD of sample mean = / sqrt(n) = 15.803 / sqrt(104) = 1.5496therefore X follows normal dist with mean = 0.5 and SD=1.5496P(X>1) = 1 - P(X