Question

Suppose x has a normal distribution with a mean of 79 and a variance of 441.00. If a sample of 15 were randomly drawn from the population, find the probability of   mu hat   for each of the following situations.

a) less than 77: probability =

b) greater than 83: probability =

c) in between 65 and 76: probability =

d) in between 76 and 94: probability =

Answer 1

Solution :

= / n = 21 / 15

(a)

P( < 77) = P(( - ) / < (77 - 79) / 21 / 15 )

= P(z < -0.37)

= 0.3557

(b)

P( > 83) = 1 - P( < 83)

= 1 - P[( - ) / < (83 - 79) / 21 / 15]

= 1 - P(z < 0.74)

= 0.2296

(c)

= P[(65 - 79) / 21 / 15 < ( - ) / < (76 - 79) / 21 / 15)]

= P(-2.58 < Z < -0.55)

= P(Z < -0.55) - P(Z < -2.58)

= 0.2862

(d)

P[(76 - 79) / 21 / 15 < ( - ) / < (94 - 79) / 21 / 15)]

= P(-0.55 < Z < 2.77)

= P(Z < 2.77) - P(Z < -0.55)

= 0.7060