In: Math
Suppose the heights of 18-year-old men are approximately normally distributed, with mean 72 inches and standard deviation 4 inches.
(a) What is the probability that an 18-year-old man
selected at random is between 71 and 73 inches tall? (Round your
answer to four decimal places.)
(b) If a random sample of thirty 18-year-old men is
selected, what is the probability that the mean height x
is between 71 and 73 inches? (Round your answer to four decimal
places.)
(c) Compare your answers to parts (a) and (b). Is the
probability in part (b) much higher? Why would you expect
this?
The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.
The probability in part (b) is much higher because the mean is larger for the x distribution.
The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.
The probability in part (b) is much higher because the mean is smaller for the x distribution.
The probability in part (b) is much higher because the standard deviation is larger for the x distribution.
Solution :
Given that ,
mean = = 72
standard deviation = = 4
P( 71< x <73 ) = P[(71-72)/4 ) < (x - ) / < (73-72) /4 ) ]
= P(-0.25 < z < 0.25 )
= P(z < 0.25 ) - P(z < -0.25 )
= 0.5793 - 0.3821 = 0.1972
Probability = 0.1972
b)
n = 30
= = 72
= / n = 4 / 30 = 0.7303
P(71< < 73 )
= P[(71-72) /0.7303 < ( - ) / < (73-72) /0.7303 )]
= P( -1.37< Z < 1.37 )
= P(Z < 1.37 ) - P(Z < -1.37 )
= 0.9066 - 0.0778
probability = 0.8288
c) The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.