Question

Women have head circumferences that are normally distributed with a mean given by mu equals 21.21 in​., and a standard deviation given by sigma equals 0.6 in. Complete parts a through c below. a. If a hat company produces​ women's hats so that they fit head circumferences between 20.4 in. and 21.4 ​in., what is the probability that a randomly selected woman will be able to fit into one of these​ hats? The probability is nothing. ​(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 nothing in. The maximum head circumference accommodated should be nothing in. ​(Round to two decimal places as​ needed.) c. If 16 women are randomly​ selected, what is the probability that their mean head circumference is between 20.4 in. and 21.4 ​in.? If this probability is​ high, does it suggest that an order of 16 hats will very likely fit each of 16 randomly selected​ women? Why or why​ not? (Assume that the hat company produces​ women's hats so that they fit head circumferences between 20.4 in. and 21.4 ​in.) The probability is nothing. ​(Round to four decimal places as​ needed.) If this probability is​ high, does it suggest that an order of 16 hats will very likely fit each of 16 randomly selected​ women? Why or why​ not? A. ​Yes, the order of 16 hats will very likely fit each of 16 randomly selected women because both 20.4 in. and 21.4 in. lie inside the range found in part​ (b). B. ​Yes, the probability that an order of 16 hats will very likely fit each of 16 randomly selected women is 0.8980. C. ​No, the hats must fit individual​ women, not the mean from 16 women. If all hats are made to fit head circumferences between 20.4 in. and 21.4 ​in., the hats​ won't fit about half of those women. D. ​No, the hats must fit individual​ women, not the mean from 16 women. If all hats are made to fit head circumferences between 20.4 in. and 21.4 ​in., the hats​ won't fit about 10.20​% of those women.

Answer 1

Solution:-

Mean = 21.21 inches, S.D = 0.60

a) The probability that a randomly selected woman will be able to fit into one of these​ hats is 0.5359.

x₁ = 20.4

x₂ = 21.4

By applying normal distribution:-

z₁ = - 1.35

z₂ = 0.317

P( - 1.35 < z < 0.317) = P(z > -1.35) - P(z > 0.317)

P( - 1.35 < z < 0.317) = 0.9115 - 0.3756

P( - 1.35 < z < 0.317) = 0.5359

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 nothing 20.141in.

p-value for the bottom 3.75% = 0.0375

z-score for the p-value = - 1.781

By applying normal distribution:-

x = 20.141

The maximum head circumference accommodated should be nothing 22.279 in.

p-value for the top 3.75% = 1 - 0.0375 = 0.9625

z-score for the p-value = 1.781

By applying normal distribution:-

x = 22.279

c) If 16 women are randomly​ selected, the probability that their mean head circumference is between 20.4 in. and 21.4 ​in. is 0.8973.

x₁ = 20.4

x₂ = 21.4

By applying normal distribution:-

z₁ = - 5.4

z₂ = 1.267

P( - 5.4 < z < 1.267) = P(z > - 5.4) - P(z > 1.267)

P( - 5.4 < z < 1.267) = 0.9999 - 0.1026

P( - 5.4 < z < 1.267) = 0.8973

B. ​Yes, the probability that an order of 16 hats will very likely fit each of 16 randomly selected women is 0.8980.