Question

A population of values has a normal distribution with μ=59.4 and σ=14.4. A random sample of size n=41 is drawn.

1. Find the probability that a single randomly selected value is between 55.4 and 62.8. ROUND ANSWER TO 4 DECIMAL PLACES!
   P(55.4<X<62.8)=
   
2. Find the probability that a sample of size n=41 is randomly selected with a mean between 55.4 and 62.8. ROUND ANSWER TO 4 DECIMAL PLACES!
   P(55.4<M<62.8)=  

Answer 1

Solution :

P( 55.4< x < 62.8) = P[(55.4 - 59.4)/ 14.4) < (x - ) /  < (62.8 - 59.4) / 14.4) ]

= P(-0.28 < z < 0.24)

= P(z < 0.24) - P(z < -0.28)

= 0.5948 - 0.3897

= 0.2051

P( 55.4< x < 62.8) = 0.2051

M = / n = 14.4 / 41 = 2.2489

P(55.4 < M < 62.8) = P[(55.4 - 59.4) / 2.2489< (M - M) / M< (62.8 - 59.4) / 2.2489)]

= P(-1.78 < Z < 1.51)

= P(Z < 1.51) - P(Z < -1.78)

= 0.9345 - 0.0375

= 0.8970  

P(55.4 < M < 62.8) = 0.8970