Suppose a simple random sample of size nequals36 is obtained from a population with mu equals 82 and sigma equals 18. (a) Describe the sampling distribution of x overbar. (b) What is Upper P left parenthesis x overbar greater than 88 right parenthesis? (c) What is Upper P left parenthesis x overbar less than or equals 76 right parenthesis? (d) What is Upper P left parenthesis 78.1 less than x overbar less than 87.55 right parenthesis?
Solution :
Given that ,
mean = = 82
standard deviation = = 18
n = 36
(a)
_{} = 82 and
_{} = / n = 18 / 36 = 18 / 6 = 3
(b)
P( > 88) = 1 - P( < 88)
= 1 - P(( - _{} ) / _{} < (88 - 82) / 3)
= 1 - P(z < 1.33)
= 1 - 0.9082
= 0.0918
Probability = 0.0918
(c)
P( < 76) = P(( - _{} ) / _{} < (76 - 82) / 3)
= P(z < -2)
= 0.0228
Probability = 0.0228
(d)
P(78.1 < < 87.55) = P((78.1 - 82) / 3 <( - _{}) / _{} < (87.55 - 82) / 3))
= P(-1.3 < Z < 1.85)
= P(Z < 1.85) - P(Z < -1.3)
= 0.9678 - 0.0968
= 0.871
Probability = 0.871
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