Question

Women have head circumferences that are normally distributed with a mean given by ​u= 23.76in, and a standard deviation given by o= 1.1in.

a. If a hat company produces​ women's hats so that they fit head circumferences between 23.1 in. and 24.1 ​in., what is the probability that a randomly selected woman will be able to fit into one of these​ hats? The probability is . 4165. ​(Round to four decimal places as​ needed.)

b. If the company wants to produce hats to fit all women except for those with the smallest ​2.5% and the largest ​2.5% head​ circumferences, what head circumferences should be​ accommodated?

The minimum head circumference accommodated should be ___in. (Round to two decimal places as​ needed.)

The maximum head circumference accommodated should be nothing ___in. ​(Round to two decimal places as​ needed.)

Answer 1

solution:-

given that mean µ = 23.76 , standard deviation σ = 1.1

a.P(23.1 < x < 24.1)

=> P((23.1-23.76)/1.1 < z < (24.1-23.76)/1.1)

=> P(-0.6 < z < 0.31)

=> P(z < 0.31) - P(z < -0.6)

=> 0.6217 - 0.2743

=> 0.3474

b. the probability is the are between the 0.025 is z = -1.96 and z = 1.96 from standard normal distribution table

from that information

formula x = z*σ + µ

=> minimum = -1.96*1.1+23.76 = 21.60

=> maximum = 1.96*1.1+23.76 = 25.92