Im not quite sure how to do this question as i cant find any

Im not quite sure how to do this question as i cant find any past work from class. Thanks in advance.

http://i.imgur.com/EgR2ter.png?1

Solution

a)

The sampling distribution is given by N(u, variance/n).

Thus,

X~N(265, 324/36), or

X~N(265, 9) [ANSWER]

This is based on the central limit theorem.

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b)

Yes, for the same reason as part a, as we know the population standard deviation.

It is thus

X~N(265,324/16) or

X~N(265,20.25) [answer]

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c)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    257      
x2 = upper bound =    261      
u = mean =    265      
n = sample size =    36      
s = standard deviation =    18      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -2.666666667      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    -1.333333333      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.003830381      
P(z < z2) =    0.09121122      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.087380839   [ANSWER]