Answer to Question #323538 in Statistics and Probability for melai

Question

The height of male college students are normally distributed with mean of 68 inches and standard deviation of 15 inches. If 25 students each are drawn from the population, what is the probability that;

2a) the sample mean is at least 75

2b) the sample mean is less than 59

2c) the sample mean is greater than 72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer 1

$a:\\P\left( \bar{x}\geqslant 75 \right) =P\left( \sqrt{25}\frac{\bar{x}-68}{15}\geqslant \sqrt{25}\frac{75-68}{15} \right) =\\=1-\varPhi \left( 2.33333 \right) =\varPhi \left( -2.33333 \right) =0.00982\\b:\\P\left( \bar{x}<59 \right) =P\left( \sqrt{25}\frac{\bar{x}-68}{15}<\sqrt{25}\frac{59-68}{15} \right) =\\=\varPhi \left( -3 \right) =0.00135\\c:\\P\left( \bar{x}>72 \right) =P\left( \sqrt{25}\frac{\bar{x}-68}{15}>\sqrt{25}\frac{72-68}{15} \right) =\\=1-\varPhi \left( 1.33333 \right) =\varPhi \left( -1.33333 \right) =0.0912$