Question

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JUUTILITIJLIULUI A population of values has a normal distribution with u = 127 and o = 30.5. You intend to draw a random sample of size n = 28. Find the probability that a single randomly selected value is between 112.6 and 140.8. P(112.6<x< 140.8) = Find the probability that a sample of size n = 28 is randomly selected with a mean between 112.6 and 140.8. P(112.6<M< 140.8) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution :

Given that μ = 127 , σ = 30.5 , n = 28

=> P(112.6 < X < 140.8) = P((112.6 - 127)/30.5 < (x - μ)/σ < (140.8 - 127)/30.5)

= P(-0.472 < Z < 0.452)

= P(Z < 0.452) − P(Z < −0.472)

= 0.6736 - 0.3192

= 0.3544

=> μx = 127 , σx = σ/sqrt(n) = 30.5/sqrt(28) = 5.7640 , n = 28

=> P(112.6 < M < 140.8) = P((112.6 - 127)/5.7640 < (M - μx)/σx < (140.8 - 127)/5.7639)

= P(-2.498 < Z < 2.394)

= P(Z < 2.394) − P(Z < −2.498)

= 0.9916 - 0.0062

= 0.9854