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

Question

Women have head circumferences that are normally distributed with a mean given by mu equals 24.74 in=24.74 in., and a standard deviation given by sigma equals 1.2 in=1.2 in. Complete parts a through c below. Women have head circumferences that are normally distributed with a mean given by H 24.74 in, and a standard deviation given by o 1.2 in. Complete parts a through c below. a. If a hat company produces women's hats so that they fit head circumferences between 24.2 in. and 25.2 in., what is the probability that a randomly selected woman will be able to fit into one of these hats? The probability is (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 3.75% and the largest 3.75% head circumferences, what head circumferences should be accommodated? The minimum head circumference accommodated should be The maximum head circumference accommodated should be (Round to two decimal places as needed.) c. If 20 women are randomly selected, what is the probability that their mean head circumference is between 24.2 in. and 25.2 in.? If this probability is high, does it suggest that an order of 20 hats wil very likely fit each ot 20 randomly selected women? Why or why not? (Assume that the hat company produces women's hats so that they fit head circumferences between 24.2 in. and 25.2 in.) The probability is (Round to four decimal places as needed.) If this probability is high, does it suggest that an order of 20 hats will very likely fit each of 20 randomly selected women? Why or why not? O A. No, the hats must fit individual women, not the mean from 20 women. If all hats are made to fit head circumferences between 24.2 in. and 25.2 in., the hats won't fit about half of those women. O B. Yes, the probability that an order of 20 hats will very likely fit each of 20 randomly selected women is 0.9342. O C. Yes, the order of 20 hats will very likely fit each of 20 randomly selected women because both 24.2 in. and 25.2 in. lie inside the range found in part (b). O D. No, the hats must fit individual women, not the mean from 20 women. If all hats are made to fit head circumferences between 24.2 in. and 25.2 in., the hats won't fit about 6.58% of those women.

Explanation / Answer

here z=(X-mean)/std deviation

a)P(24.2<X<25.2)=P(-0.45<Z<0.3833)=0.6493-0.3264=0.3229

b)for upper and lower 3.75%; corresponding z=+/-1.7805

hence minimum =mean-z*std deviation=26.88

maximum=mean+z*std deviation=22.60

c) std error =std deviaiton/(n)1/2 =0.2683

hence P(24.2<X<25.2)=P(-2.0125<Z<1.7143)=0.9568-0.0221=0.9342

option A is correct

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