consider a population that is decribed by a random variable

consider a population that is decribed by a random variable that takes the values 0.4 and 3.4 with probabilities 0.8 and 0.2, respectively.

1) if a single sample is drawn at random from this population , find the probability that its values will exceed 1.1

2) if two samples are drawn independently at random from this population find the (exact) probability that their average value will exceed 1.1

3) if 225 samples are drawn independently at random from this population, estimate the probability that their aaverage value will exceed 1.1

Solution

1.

It has to be 3.4 in this case, so

P(x>1.1) = 0.2 [ANSWER]

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2.

So the average exceeds 1.1, there should be at least one 3.4. That is, it cannot be both 0.4.

Thus,

P(Xbar > 1.1) = 1 - 0.8^2 = 0.36 [ANSWER]

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3.

Thus,  
  
E(x) = Expected value = mean =    0.92
Var(x) = E(x^2) - E(x)^2 =    1.5376
s(x) = sqrt [Var(x)] =    1.24

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    1.1      
u = mean =    0.92      
n = sample size =    225      
s = standard deviation =    1.24      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2.177419355      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.177419355   ) =    0.014724646 [ANSWER]