Suppose the sediment density (g/cm) of a randomly selected specimen from a certa

Question

Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.6 and standard deviation 0.70. (a) If a random sample of 25 specimens is selected, what is the probability that the sample average sediment dtndey 2 at te 2 nd 3007 ur dcinal plbca) most 3.00 between 2.6 and 3.00 (b) How large a sample size would be required to ensure that the probability in part (a) is at least 0.99? (Round your answer up to the nearest whole number.) specimens You may need to use the appropriate table in the Appendix of Tables to answer this question

Explanation / Answer

a. Here the population standard deviation is known so we can use normal(z) distribution.

If x is the random variable denotes the density

p(x<=3)=p(z<=(3-2.6)/(0.70/sqrt(25)))= p(z<=2.857143)=pnorm(2.857143)=0.9979

p(2.6<=x<=3)=p(0<=z<=2.857143)=pnorm(2.857143)-pnorm(0)=0.9979-0.5=0.4979

b. here we need to find n such that pnorm(z)>0.99

or z>qnorm(0.99)=2.326348 or (3-2.6)/(0.70/sqrt(n)) >=2.326348 or sqrt(n)>= 2.326348/0.514286 =4.523452

or n>=4.523452**2=20.46162

so n>=21 (next whole number)

So we need at least 21 samples such that probability of part a will be at lease 0.99

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