Answer to Question #316716 in Statistics and Probability for hoon

Question

The height of grade 1 pupils are approximately normally distributed with µ = 45 inches and s = 2.

• If an individual pupil is selected at random, what is the probability that he or she has a height of 42 and 47?
• A class of 30 of these pupils is used as a sample. What is the probability that the class mean is between 42 and 47?
• If a pupil is selected at random, what is the probability that is taller than 46 inches?
• A class of 30 of these pupils is used as sample. What is the probability that the class mean is greater than 46 inches?

Answer 1

$i:\\P\left( 42<X<47 \right) =P\left( \frac{42-45}{2}<\frac{X-45}{2}<\frac{47-45}{2} \right) =\\=P\left( -1.5<Z<1 \right) =\varPhi \left( 1 \right) -\varPhi \left( -1.5 \right) =0.84135-0.06681=0.77454\\ii:\\P\left( 42<\bar{x}<47 \right) =P\left( \sqrt{30}\frac{42-45}{2}<\sqrt{30}\frac{\bar{x}-45}{2}<\sqrt{30}\frac{47-45}{2} \right) =\\=P\left( -8.21584<Z<5.47723 \right) =\varPhi \left( 5.47723 \right) -\varPhi \left( -8.21584 \right) =\\=1-2.2\cdot 10^{-8}-1.1\cdot 10^{-16}\approx 1\\iii:\\P\left( X>46 \right) =P\left( \frac{X-45}{2}>\frac{46-45}{2} \right) =P\left( Z>0.5 \right) =\\=\varPhi \left( -0.5 \right) =0.3085\\iv:\\P\left( \bar{x}>46 \right) =P\left( \sqrt{30}\frac{\bar{x}-45}{2}>\sqrt{30}\frac{46-45}{2} \right) =P\left( Z>2.7386 \right) =\\=\varPhi \left( -2.7386 \right) =0.00309$