Women have head circumferences that are normally distributed

Women have head circumferences that are normally distributed with a mean given by

mu equals 24.66 in=24.66 in.,

and a standard deviation given by

sigma equals 0.8 in=0.8 in.

Complete parts a through c below.

If 11women are randomly selected, what is the probability that their mean head circumference is between

24.6

in. and

25.6

in.? If this probability is high, does it suggest that an order of

1111

hats will very likely fit each of

1111

randomly selected women? Why or why not? (Assume that the hat company produces women\'s hats so that they fit head circumferences between

24.624.6

in. and

25.625.6

in.

Solution

Mean ( u ) =24.6
Standard Deviation ( sd )=0.8
Number ( n ) = 11
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a)
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 24.6) = (24.6-24.6)/0.8/ Sqrt ( 11 )
= 0/0.2412
= 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(X < 25.6) = (25.6-24.6)/0.8/ Sqrt ( 11 )
= 1/0.2412 = 4.1458
= P ( Z <4.1458) From Standard Normal Table
= 0.99998
P(24.6 < X < 25.6) = 0.99998-0.5 = 0.5                  

No, since the probability is 0.50, probability
figured is half only, hat company produces women\'s hats
are does n\'t likely to fit each of women