Let the random variable X follow a Normal distribution with mean = 50 and varian

Question

Let the random variable X follow a Normal distribution with mean = 50 and variance 2-64. a. Find the probability that X is greater than 60. b. Find the probability that X is greater than 35 and less than 62. c. Find the probability that X is less than 55 d. The probability is 0.2 that X is greater than what number? Basic Exercise 6.6 (p.242) Given a population with a mean of -100 and a variance of 2-900, the central limit theorem applies when the sample size is n 25. A random sample of size n = 30 is obtained. a. What are the mean and variance of the sampling distribution for the sample means? b. What is the probability that > 109? c. What is the probability that 96 sis 110? d. What is the probability that a s 107?

Explanation / Answer

here std deviation =(64)1/2 =8

a) P(X>60)=1-P(X<60)=1-P(Z<(60-50)/8)=1-P(Z<1.25)=1-0.8944 =0.1056

b) P(35<X<62)=P(-1.875<Z<1.5)=0.9332-0.0304 =0.9028

c)P(X<55)=P(Z<0.625)=0.7340

d) for top 20% values; at 80th percentile ; z=0.8416

hence corresponding value =mean+z*std deviaiton =50+0.8416*8=56.73

6.6)

a)mean =100

variance =900/30=30

b)for std deviaiton=(30)1/2 =5.477

hence P(Xbar>109)=1-P(Xbar<109)=1-P(Z<(109-100)/5.477)=1-P(Z<1.6432)=1-0.9498 =0.0502

c)P(96<Xbar<110)=P(-0.7303<Z<1.8257)=0.9661-0.2326 =0.7335

d)P(Xbar<107)=P(Z<1.2780)=0.8994

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