A A population of values has a normal distribution with 2085

A. A population of values has a normal distribution with =208.5 and =35.4. You intend to draw a random sample of size n=236.

Find the probability that a single randomly selected value is greater than 203.4.
P(X > 203.4) =  Round to 4 decimal places.

Find the probability that the sample mean is greater than 203.4.
P(X¯¯¯ > 203.4) =  Round to 4 decimal places.

B. A population of values has a normal distribution with =223.7 and =56.9. You intend to draw a random sample of size n=244.

Find the probability that a single randomly selected value is between 217.5 and 234.6.
P(217.5 < X < 234.6) =  Round to 4 decimal places.

Find the probability that the sample mean is between 217.5 and 234.6.

P(217.5 < X¯¯¯ < 234.6) =  Round to 4 decimal places.

Solution

A. A population of values has a normal distribution with =208.5 and =35.4. You intend to draw a random sample of size n=236.

Find the probability that a single randomly selected value is greater than 203.4.
P(X > 203.4) =  Round to 4 decimal places.
z value for 203.4, z=(203.4-208.5)/35.4   = -0.14

P( X > 203.4) = P(z > -0.14)

= 0.5557

Find the probability that the sample mean is greater than 203.4.
P(X¯¯¯ > 203.4) =  Round to 4 decimal places.

Standard error = sd/sqrt(n) =35.4/sqrt(236) =20.0377

z value for 203.4, z=(203.4-208.5)/20.0377 = -0.25

P( mean X > 203.4) = P(z > -0.25)

= 0.5987

B. A population of values has a normal distribution with =223.7 and =56.9. You intend to draw a random sample of size n=244.

Find the probability that a single randomly selected value is between 217.5 and 234.6.
P(217.5 < X < 234.6) =  Round to 4 decimal places.

z value for 217.5, z=(217.5-223.7)/56.9 = -0.11

z value for 234.6, z=(234.6-223.7)/56.9 = 0.19

P(217.5 < X < 234.6) = P( -0.11<z<0.19)

=P( z < 0.19) –P( z <-0.11)

= 0.5753 - 0.4562

=0.1191

Find the probability that the sample mean is between 217.5 and 234.6.

Standard error = sd/sqrt(n) =56.9/sqrt(244) =3.6426

z value for 217.5, z=(217.5-223.7)/3.6426 = -2.25

z value for 234.6, z=(234.6-223.7)/3.6426 = 2.99

P(217.5 < meanX < 234.6) = P( -2.25<z<2.99)

=P( z < 2.99) –P( z <-2.25)

= 0.9986 - 0.0122

= 0.9864