A simple random sample of size n equals =10 is obtained from a population with μ equals = 62 and σ equals = 18. (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x overbar x. (b) Assuming the normal model can be used, determine P( x overbar x less than < 65.7). (c) Assuming the normal model can be used, determine P( x overbar x greater than or equals ≥ 63.6).
Solution :
Given that ,
mean = = 62
standard deviation = = 18
n = 10
a) _{} = = 62
_{} = / n = 18 / 10 = 5.69
b) P( < 65.7) = P(( - _{} ) / _{} < ( 65.7 - 62) / 5.69 )
= P(z < 0.65)
Using z table
= 0.7422
c) P( 63.6 ) = 1 - P( 63.6 )
= 1 - P[( - _{} ) / _{} ( 63.6 ) / 5.69 ]
= 1 - P(z 0.28 )
Using z table
= 1 - 0.6103
= 0.3897
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