Question

# A simple random sample of size n equals =10 is obtained from a population with μ...

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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