Question

Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. (a) What is the probability that a randomly chosen 10 year old is shorter than 51 inches? (Round your answer to four decimal places.) (b) What is the probability that a randomly chosen 10 year old is between 57 and 67 inches? (Round your answer to four decimal places.) (c) If the tallest 10% of the class is considered "very tall", what is the height cutoff for "very tall"? (Round your answer to two decimal places.) in (d) The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What percent of 10 year olds cannot go on this ride? (Round your answer to two decimal places.)

Answer 1

`\mu = 55`

`\sigma = 6`

(a) What is the probability that a randomly chosen 10 year old is shorter than 51 inches?

Formula :
`z=(x-\mu)/(\sigma)`

`z=(51-55)/(6)`

`z=−0.666`

refer the z table

P(z<-0.66)=0.2546

Hence the probability that a randomly chosen 10 year old is shorter than 51 inches is 0.2546

(b) What is the probability that a randomly chosen 10 year old is between 57 and 67 inches?

Formula :
`z=(x-\mu)/(\sigma)`

`z=(57-55)/(6)`

`z=0.333`

refer the z table

P(z<0.333)=0.6293

`z=(67-55)/(6)`

`z=2`

refer the z table

P(z<2)=0.9772

P(57<x<67)=P(0.333<z<2)=P(z<2)-P(z<0.333)= 0.9772-0.6293=0.3479

Hence the probability that a randomly chosen 10 year old is between 57 and 67 inches is 0.3479

c) If the tallest 10% of the class is considered "very tall", what is the height cutoff for "very tall"?

1 - 0.10 = 0.9

Refer the z table

probability of shortest = 0.98

z corresponding 0.9 = 1.28

`1.28=(x-55)/(6)`

`7.68=x-55`

`7.68+55=x`

`62.68=x`

Hence the height cutoff for "very tall" is 62.98

d)The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What percent of 10 year olds cannot go on this ride?

`z=(54-55)/(6)`

`z=−0.166`

p(z<-0.166)=0.4364

Hence 43.64% of 10 year olds cannot go on this ride